Talk:Mathematical coincidence

Another coincidence
Forgive the intrusion by someone who is not a proper mathematician (an economist). Here's one that may have been included-then-deleted: 2 approx= exp((1/2)^(1/2))

Not that good an approximation (rhs=2.028, and I don't know what approximation it arises from) but it seems to me to be rather pleasing. — Preceding unsigned comment added by Stephen Wright Birkbeck (talk • contribs) 18:16, 5 January 2014 (UTC)

Error in the page
[]

The fifth expression in "Decimals" should evaluate to 37, not 370.

Name of article
A few comments. Maybe mathematical coincidence might yet be a better page name. I'm not sure what a 'theoretical explanation' would be, in all cases: but certainly in the case of exp(π√163) there is a very good if hidden reason. Perhaps the pigeonhole principle could be invoked. After all, it is because of Dirichlet's theorem on diophantine approximation, based on it, that we know that good rational approximations exist. And so on, for other kinds of relations. By the way, I added the law of small numbers link, but that really needs disambiguation since the Poisson distribution meaning is a separate thing.

Charles Matthews 16:10, 21 Jun 2004 (UTC)

Hi Charles

you're probably right about the page name. I chose the name originally because I reckoned that there would be hundreds and hundreds. In the end, the best I could do wasn't very extensive! There must be more out there! (exactly the same thing happened with List of scientific howlers in literature)


 * The scientific howlers list is likely to grow, slowly but surely. Dpbsmith 22:58, 21 Jun 2004 (UTC)

- Anyway, the exact definition of a mathematical coincidence is problematical. I spent some considerable time pondering the best definition, only to get myself tangled up in philosophy, and at one point defining everything to be a coincidence. Of course $$\pi^2\simeq 10$$! How could it possibly be otherwise? Nevertheless, there is definitely _something_ that all the examples have in common. Except maybe the exp(pi*sqrt(163)) one, which as you point out is a result of some algebraic number theory (all of which I forgot immediately after my finals). And possibly the continued fraction ones, although it _is_ coincidental that the continued fraction for pi has a large coefficient early on (isn't it?)

Best wishes

Robinh 20:51, 21 Jun 2004 (UTC)

From a concerned mathematician
I am a mathematician who studies (among other things) topological (Nielsen) coincidence theory. The word "coincidence" has a widely accepted and recognized technical meaning: given two functions f and g, the Coincidence Set Coin(f,g) is the set of points x such that f(x) = g(x).

This is important as it is (perhaps) the most natural generalization of fixed point (mathematics) theory (where g is taken to be the identity map) and is the original setting for many results (e.g. the Lefschetz fixed-point theorem) which are now commonly misidentified as being fixed point results. (Lefschetz proved first the coincidence version of his result, and noted the fixed point version as a specific case.)

Coincidence theory also has an important reduction when the function g is taken to be a constant map- this setting is called root theory (finding points x with f(x) = c for some constant c).

As a mathematician, I feel like a wikipedia page on coincidences should refer to the above concept. The content now described as "mathematical coincidence" would perhaps be more accurately described as "mathematical curiosity".

My $0.02. Chris Staecker, UCLA Math Dept


 * Well, we can have a page coincidence set, or coincidence (mathematics), any time you want. Charles Matthews 19:30, 26 September 2005 (UTC)


 * If you ever decide to make those pages, I trust no one would object to the placement of a link to it at the top of the page instructing people on ambiguity of the title.Wrath0fb0b 14:26, 16 March 2006 (UTC)

Two levels of "coincidence"
It seems to me that there's an important distinction between "coincidences" that arise out of a general theory, like the one about exp(pi sqrt 163), and ones for which no explanation of any kind is known. To take a few examples from the page as of 2005-04-10: Is it worth making this distinction on the page? Gareth McCaughan 13:48, 2005 Apr 10 (UTC)
 * e^pi ~= pi^e is true because pi is close to e and x / log x is stationary at e. Not much of a coincidence, really; the same would be just as true if 3 or sqrt(10) were used in place of pi.
 * pi ~= 355/113 is a "real" coincidence, so far as I know: that is, no reason is known why pi should have so large a coefficient so early in its continued fraction.
 * sqrt(2) ~= 17/12 is not much of a coincidence; there's a general theorem that says that roots of quadratic equations always have rational approximations that are at least about that good.
 * exp(pi) ~= pi+20 seems to be a "real" coincidence.
 * 1 mile / 1 km ~= phi is an absolute coincidence, if anything is. (Incidentally, it might be worth mentioning the Zeckendorff representation in connection with this...)


 * You make an excellent point. One might as well claim that "pi ~= 3.14" is a coincidence. How about pi^4 + pi^5 ~= e^6? MrHumperdink 16:24, 7 May 2006 (UTC)
 * Cooool... I'm adding this to the article.--Army1987 18:59, 7 May 2006 (UTC)
 * It is cool, but many of these are not. pi ~= 22/7 is not a coincidence, it is an approximation. I'm going to be bold and go through the list, removing what do not seem to be coincidences. Obviously my point of view should be rechecked by other editors; I'll list what I take out here. MrHumperdink 00:31, 8 May 2006 (UTC)

The title of this page
How about "numerical coincidences"? Gareth McCaughan 13:48, 2005 Apr 10 (UTC)

these are not coincidences
None of these can be classified as coincidences, except maybe
 * $$e^\pi\simeq\pi^e$$. -Hmib 02:48, 20 Jun 2005 (UTC)
 * which is explained by the fact that e and pi are both close to 3. --MarSch 13:39, 28 October 2005 (UTC)

Not-Coincidences Removed
I have removed the following items from the list:

''Up to here, all of these are simply approximations. So pi is close to 3 and slightly greater... obviously if you square it you get close to 10. Similarly, one might as well call it a "coincidence" that pi ~= 314/100, as to consider 22/7 and and the other fractions.''
 * $$ \sqrt{2 \pi} \simeq 5/2 $$ to about 0.1% (one part in a thousand).
 * $$\pi\simeq 22/7$$; correct to about 0.04%; $$\pi\simeq 355/113$$, correct to six places or 0.000008%.
 * $$\pi^3\simeq31$$; correct to about 0.02%.
 * $$\pi^2\simeq10$$; correct to about 1.3%. This coincidence was used in the design of slide rules, where the "folded" scales are folded on $$\pi$$ rather than $$\sqrt{10}$$, because it is a more useful number and has the effect of folding the scales in about the same place; $$\pi^2\simeq 227/23$$, correct to 0.0004% (note 2, 227, and 23 are Chen primes).
 * $$\pi^4\simeq 2143/22$$, accurate to about one part in $$10^{10}$$; due to Srinivasa Ramanujan, who might have noticed that the continued fraction representation for $$\pi^4$$ begins $$[97; 2,2,3,1,16539,1,1,\ldots]$$. See also: Pi culture.
 * $$\pi^5\simeq306$$; correct to about 0.006%.

Continued:


 * $$1+1/\log(10)\simeq 1/\log(2)$$; leading to Donald Knuth's observation that, to within about 5%, $$\log_2(x)=\log(x)+\log_{10}(x)$$.
 * This one I'm unsure about. I took it out because it seems too complex or contrived to qualify.
 * $$\pi$$ seconds is a nanocentury (ie $$10^{-7}$$ years); correct to within about 0.5%
 * Yes, or 3, or 3.1, or... basically, nothing special about pi here.
 * one attoparsec per microfortnight approximately equals 1 inch per second (the actual figure is about 1.0043 inch per second).
 * "Microfortnight"? No.
 * a cubic attoparsec (a cube where each edge is one attoparsec) is within 1% of a fluid ounce.
 * Again, what's special about an attoparsec, or a microfortnight, or whatever?
 * $$\pi\simeq\frac{63}{25}\left(\frac{17+15\sqrt{5}}{7+15\sqrt{5}}\right)$$; accurate to 9 decimal places (due to Ramanujan).
 * What an astounding coincidence.
 * NA ≈ 279, where N is Avogadro's number; correct to about 0.4%. This means that a yobibyte is slightly more than two moles of bytes.
 * The speed of light is about one foot per nanosecond (accurate to 2%) or 3&times;108 m/s (accurate to about 0.1%)

Well, that's it for now. I tried to look for similarities which were neither too simple (e + 0.4 is close to pi! Stop the presses!) or too complex (see Ramanujan's approximation of pi). I hope this helps... MrHumperdink 01:08, 8 May 2006 (UTC)

reversion of mass delete
Hi. You removed 13 points. I disagree with 10 of these deletions (which I consider to be valid entries), and am unsure about the other 3. Perhaps a good way forward would be to delete entries one at a time, and then we discuss each one on its merits.

best wishes, Robinh 07:23, 8 May 2006 (UTC)
 * Please, tell me why you consider these to be coincidences, rather than simply reverting my edit in its entirety. I have discussed each one's merits. It is your place to respond now. Approximations are not "coincidences". They can be explained. That pi^4 + pi^5 ~= e^6 is pretty hard to explain, and is relatively simple. And so on. MrHumperdink 14:50, 8 May 2006 (UTC)


 * I am for removal of approximations - they are not coincidences. What so special about pi = 22/7? Is 22 somewhat special, or 7? Better mention some (not really coincidences, but amazing properties of numbers) other things, like friendly numbers: 220 equals sum of all (proper) divisors of 284 (1+2+4+71+142), while 284 equals the sum of all (proper) divisors of 220 (1+2+4+5+10+11+20+22+44+55+110). --Alextalk 15:55, 8 May 2006 (UTC)
 * That makes two of us, then. Robinh, please state your reasons for reverting my edit. I won't start an edit war - no worries there :) - but I would like to continue the discussion. I'll leave you a message. MrHumperdink 02:20, 9 May 2006 (UTC)
 * The somewhat special thing is that pi has such a large term as 292 very early in its continued fraction, and that the first powers thereof all have big numbers quite early in their CFs, in the case of π^3 and π^5 as soon as the first denominator. --Army1987 19:58, 10 May 2006 (UTC)

Also note that in the first entry of attoparsec we can replace it with picoparsec, and remove the micro from microfortnight. Alfa-ketosav (talk) 05:07, 30 August 2019 (UTC)

Scope of page
This discussion and the page itself shows there is some uncertainty about the scope of this page. I can't really see there's any difference between a co-incidence and an approximation, except the former must at least seem slightly surprising in its closeness. The lack of a theoretical explanation is demanded in the introduction, yet we have the $$e^{\pi\sqrt{n}}$$ co-incidences included (possibly my favourite example). I suggest the lack of a theoretical explanation comment should be dropped, as its only effect would be to remove the most interesting examples.

Also, I would like to include "co-incidental" equalities, such as:
 * $$\sum_{n=1}^{24} n^2 = 70^2\;$$

a remarkable, unique and highly significant "co-incidence".

Elroch 22:06, 8 May 2006 (UTC)


 * Exactly my point. $$e^{\pi\sqrt{n}}$$ clearly should be removed (log(n+14) also passes near integers especially when n=163. So?). The mile being phi kilometers is quite a coincedence. The number pi as irrational number being well approximated by a member of "everywhere dense" subset of rational numbers (for example 22/7) is not. I would like to ask Robinh to remove all examples of "theoretically obvious" approximations. PLEASE. --Alextalk 05:09, 9 May 2006 (UTC)

Coincidences or approximations
Hello everyone.

sorry to revert a good-faith edit. (No problem. MrHumperdink 00:44, 10 May 2006 (UTC)) Well, first of all why not split the page into mathematical coincidence and mathematical approximation? This would have the advantage that $$\pi^2\simeq 10$$ (which is certainly notable from a slide rule perspective) would have a place in the encyclopedia. Would that be a good idea?

(Let's discuss the others when we've come to a conclusion on the pi ones)

On the other hand, I do feel that $$\pi\simeq 22/7$$ is a coincidence as well as an approximation. The interesting fact is that the second quotient in the continued fraction representation of pi (15) is large. There seems to be no explanation for this (interesting, notable) fact than "the CF expansion is what it is".


 * $$\pi\approx 22/7$$ is an approximation proven by Archimedes through calculating perimeter of a circumscribing 96-sided proper polygon. No real coincidences here. --Alextalk 15:40, 9 May 2006 (UTC)

Also, taking $$\pi^4\simeq 2143/22$$ and the other longer identity for pi. Now, Ramanujan himself considered these to be notable coincidences (although I must confess that I can't put my finger on an original source). Ramanujan was a genius and if he thought that a near-equality was notable enough to write down, then it's certainly notable enough for wikipedia.

best wishes, Robinh 07:21, 9 May 2006 (UTC)


 * 1. On "Mathematical approximation" article. Not very good idea, at least with this name. There exist Approximation, Diophantine approximation, Thue–Siegel–Roth theorem to name a few on the topic.
 * 2. While on the subject. I wonder why are you using symbol $$\simeq$$ instead of $$\approx$$?
 * 3. $$\pi^2\approx 10$$ - is a coincidence, if you consider that an area of a unit circle has nothing to do with the number of fingers. But 22/7 is still an approximation and nothing more. 22 and 7 - some regular numbers, not representing anything else. I don't see significant coincidence: nothing cool about it.
 * 4. We should take each approximate identity one-by-one and discuss it (with arguments, of course). Some may definitely survive, the rest should be removed. I agree that massive deletes are not a way to go, but some clenup is needed.
 * 5. We can also take another approach, let participants rank "coincidences" by a level of "amusement" and absense of theoretical explanation. And lets kill the least "amusing" ones - those which are kind of expected and uninteresting. Surely, Ramanujan's long near identity for pi would be high on my list.
 * --Alextalk 13:58, 9 May 2006 (UTC)


 * How about few simple ones:
 * $$\frac {16} {64} = \frac {1\not6} {\not6 4} = \frac {1} {4}$$,    $$\frac {26} {65} = \frac {2\not6} {\not6  5} = \frac {2} {5}$$,    $$\frac {19} {95} = \frac {1\not9} {\not9  5} = \frac {1} {5}$$ :)--Alextalk 14:55, 9 May 2006 (UTC)
 * First: clever. Nice fractions there... :P I can agree about $$\pi^2\approx 10$$ now that I understand its truly coincidental quality. I still really disagree with 22/7 and even 2143/22. As I understand it, a "coincidence" is when two independent concepts (or events or whatever) suddenly meet. As Alex said, 22 and 7 are not "special" numbers. It's not enough to be unexpected (the birthday paradox, for instance, is "unexpected" but certainly not a "coincidence" by any meaning of the word); there must be two separate ideas put together. For instance, $$\pi^4 + \pi^5\approx e^6$$ is coincidental because it links two "special" and unconnected numbers unexpectedly. I really don't consider Ramanujan's approximation of pi. 63, 25 and 15sqrt(5)? How is it a coincidental expression? etc etc. MrHumperdink 00:44, 10 May 2006 (UTC)

Approx vs. Simeq notation
First things first. Symbol \Approx ($$\approx$$) means approximately equal and should be used when dealing with numbers (for example $$\pi \approx 3$$). Symbol \Simeq ($$ \simeq $$) on the other hand means similar or equal, which is definitely not the case with fixed values like numbers, it is usually used with functions (for example $$\sin x \simeq x$$). I believe I am clear on this: all number relationships should use $$\approx$$ instead of $$\simeq$$. When we will get to assymptotic approximations/coincidences for functions then we will use $$ \simeq $$. Ok? --Alextalk 03:09, 10 May 2006 (UTC)


 * no, not okay. $$\approx$$ means approximately equal and there is no reason why it couldn't be used for functions. $$ \simeq $$ means homotopic.--MarSch 15:02, 11 May 2006 (UTC)


 * You better search Wikipedia for misuse of $$ \simeq $$ then. Because I found $$\sin x \simeq x$$ in one of the articles. --Alextalk 17:13, 11 May 2006 (UTC)

Ranking
Lets rank article examples to determine least impressive and highly anticipated with the goal to remove them. Here are my choices. I consider highly non-coincidental (candidates for deletion) the following examples: $$\pi^7 \approx 3020$$ with precision of 0.01%? You can continue to infinity, just multiply by $$\approx 10$$. Removal of these non-coincidental examples would be a good start. --Alextalk 03:29, 10 May 2006 (UTC)
 * $$\pi^3 \approx 31$$. What did you expect if $$\pi^2 \approx 10$$? 43? 55? Of course, it's almost 31.
 * $$\pi^5 \approx 306$$. My six-years old knows that if $$\pi^2 \approx 10$$ and $$\pi^3 \approx 31$$ then $$\pi^5 \approx 306$$. How about
 * $$\pi \approx 22/7$$. Every schoolboy in ancient Egypt knew this one - it kind of lost its element of surprise (if it ever had any).
 * pi^7 = 3020.2932.. is not very impressive. pi^3 = 31.006... definitely is. pi^5 = 306.019... probably is. --Army1987 18:36, 11 May 2006 (UTC)
 * pi^7 is no more or less impressive than pi^2 = 9.8696... and yet we include "pi^2 ~= 10".--MrHumperdink 18:07, 12 May 2006 (UTC)
 * I agree that, from a purely mathematical POV, pi^2 = ~10 is not very impressive. But I guess it is here because it's one of the approximate identities which have a practical usefulness. (However, I have no opinion on wheter to remove it from here.) --Army1987 13:13, 13 May 2006 (UTC)

Hi.

I think you're right about the pi powers. Trying the same thing with exp(n) gives similarly (un)surprising results.

But I'm unsure about $$\pi\approx 22/7$$. The surprising thing about this is not the 22 nor the 7 per se, but the notable and interesting fact that in this close approximation, the "7" is small, due to the large quotient in pi's continued fraction expansion (15) occurring unusually early. OTOH, trying it with "random" nonquadratic irrationals gives a similar story. For example, $$\pi+e$$ (e itself is a bad idea as its CF representation is semi-regular) comes out with 41/7, which is about as good an approximation as 22/7 for pi.

Robinh 07:58, 10 May 2006 (UTC)


 * I'll give you this: $$\pi\approx 22/7$$ is somewhat of a coincidence, but it is not very impressive if it requires a paragraph of explanations. I change my position from "definitely remove" to "borderline boring". You can keep it, but don't stop looking for more interesting things. --Alextalk 15:05, 10 May 2006 (UTC)


 * It doesn't matter if the approximation is "a notable and interesting fact" but rather if it is coincidental. Ditch the approximations. I understand that the large denominators are unusual, but you have yet to show how they are part of an actual coincidence. --MrHumperdink 22:47, 10 May 2006 (UTC)


 * Robinh, the problem with approximations is that there is no feeling of uniqueness. You take some irrational number with large quotient and approximate it with the small denominator fraction claiming that it's close approximation. How rare is that? You mentioned $$\pi+e$$ has as good approximation as $$\pi$$, so I assume all irrational numbers which differ from $$\pi$$ somewhere beyound 100th digit have the same property. Nothing really special here. Nor beautiful. --Alextalk 04:08, 11 May 2006 (UTC)


 * Yes, that was my point: many irrationals that crop up in this line have similarly good approximations with similarly small denominators.  Best, Robinh 07:03, 11 May 2006 (UTC)


 * If you want interesting fact about approximations, here it is: Golden ratio is the "most irrational" number in a sense that it has the slowest convergence of rationals with increasing denominators approximating it. No coincidence here, because $$\phi$$ is an infinite chain fraction of 1s
 * $$1 + \frac{1} {1 + \frac{1} {1 + \frac{1} {...}}}$$ --Alextalk 04:08, 11 May 2006 (UTC)


 * So... since you've agreed, Robinh, that the approximations have similar fractions, then there is nothing noteworthy about the powers of pi. I will wait a day, then remove them. --MrHumperdink 18:07, 12 May 2006 (UTC)

After a day, I removed the following five approximations:
 * 1) $$ \sqrt{2 \pi} \approx 5/2 $$ to about 0.1% (one part in a thousand).
 * 2) $$\pi\approx 22/7$$; correct to about 0.04%; $$\pi\approx 355/113$$, correct to six places or 0.000008%.
 * 3) $$\pi^3\approx31$$; correct to about 0.02%.
 * 4) $$\pi^4\approx 2143/22$$, accurate to about one part in $$10^{10}$$; due to Srinivasa Ramanujan, who might have noticed that the continued fraction representation for $$\pi^4$$ begins $$[97; 2,2,3,1,16539,1,1,\ldots]$$. See also: Pi culture.
 * 5) $$\pi^5\approx306$$; correct to about 0.006%.
 * (The theory of continued fractions gives a systematic treatment of this type of coincidence; and also such coincidences as $$2\times 12^2\approx 17^2$$ (ie $$\sqrt{2}\approx 17/12$$). Curiously the continued fractions of the first few powers of $$\pi$$ have big numbers (>50) quite early, in the case of $$\pi^3$$ and $$\pi^5$$ as soon as the first denominator.) --MrHumperdink 21:25, 13 May 2006 (UTC)


 * More on continued fractions. Periodic continued fractions (which are quadratic irrationals) explain and estimate the goodness of Ramanujan approximation $$\pi\approx\frac{63}{25}\left(\frac{17+15\sqrt{5}}{7+15\sqrt{5}}\right)$$. Perhaps, in his time it was hard to discover things like that, but now well-studied periodic continued fractions allow to produce any number similar expressions. --Alextalk 06:32, 14 May 2006 (UTC)


 * So, I propose waiting a day again, in case Robinh has a comment. Then we can remove that one. --MrHumperdink 17:39, 14 May 2006 (UTC)
 * That one looks very artificial, I agree with removing it. Anyway: 1) I don't understand why the ones above were removed but pi^2 = 227/23 remains; 2) IMO that ones should be re-added, especially the one about pi^4, such a large term so early in its CF is unexpectable. --Army1987 19:42, 14 May 2006 (UTC)


 * Hi; thanks for waiting. You guys gonna love this.  I've found the original reference in a paper of Ramanujan's of 1914.  The expression is the twenty-fifth in a series of increasingly accurate approximations to pi.  Ramanujan states, and I quote, "we cannot expect a high degree of approximation for small values of $$n$$".  So I vote delete (and indeed I'm gonna delete it myself, on the grounds that noone else has ever expressed any support for keeping it :-)


 * Now, there is an interesting outcome too. The next section in the same paper states, and I quote directly, "Another curious approximation to pi is $$\left(9^2+\frac{19^2}{22}\right)^{1/4}$$.  This value was obtained empirically, and it has no connection with the preceding theory".  This approximation is equivalent to $$\pi^4=2143/22$$, which I'll re-add to the article, with a reference.    Best wishes, Robinh 07:22, 15 May 2006 (UTC)
 * That one can be stated another way: write 1234, swap the first two digits, swap the last two digits, divide by two-two, take the 2nd (square) root two times, and you get approximately pi. (That one was on the article pi a long time ago.) --Army1987 15:11, 15 May 2006 (UTC)

Other curiosities
Added section on amusing equalities. Not sure it belongs to this article, but it is fun reading. --Alextalk 15:36, 10 May 2006 (UTC)


 * This new section seems closest to what the article is getting at: unexpected equalities. I especially liked (1+9+6+8+3)^3=19683. That is a coincidence, I think. And, as you noted, they are actually interesting to hear.

Please, can we remove the Ramanujan one I keep mentioning? Or at least explain why it is coincidental? --MrHumperdink 22:47, 10 May 2006 (UTC)

Defining coincidence
I'm still rather unsure about the boundary between a coincidence which has an "explanation" and one which does not. dictionary.com defines coincidence as "The state or fact of occupying the same relative position or area in space", which seems more appropriate than the alternative definition about chance events. No mathematical coincidence is a chance event, since they are all determined precisely by the underlying axioms, with no uncertainty. I am sure that for any apparently "chance" co-incidence, someone could come up with an indirect argument why the approximation was as good as it is, rather than just doing the calculation and seeing that it is. Just because an approximation is "explained" by a large term in a continued fraction expansion is not a reason to exclude it, since good approximations will obviously tend to be associated with such continued fractions.

There should be some role for information theory here - a coincidence could be defined as an unusually compressed form for a good approximation. For example, a certain amount of information is needed for a fraction, and this amount of information can be compared to the error in an approximation. (It may be best to just compare the information content of the denominator to the accuracy) Elroch 11:36, 15 May 2006 (UTC)
 * Obviously all approximate equalities are determined and chance plays no role. But we could define a coincidence as an unexpected approximate equality. Yes, in the case of CFs with large terms, coincidences do have a reason, but an unexpected one. Nobody would expect the CF of pi^4 to have a five-digit term so early before calculating (or reading about) it. In the case of approximation of irrational number by a fraction, we could for example define a coincidence as when a term in the CF is much bigger than the denominator of the fraction obtained with the terms before it. For example, IIRC, pi = [3, 7, 15, 1, 292, 1...]; fifteen is more than twice as much as the denominator in 22/7, and two hundred and ninety-two is more than twice the denominator of 355/113. (Of course, this way many approximate equalities with integers could be deemed coincidences; but we could say that this shouldn't apply to integers, and approximate equalities with integers should be considered coincidences only when a number is within 0.02 of an integer.)

Anyway, I'm removing pi^e = e^pi. That's definitely not a coincidence, it happens with any two numbers enough close to e. For example, 2.5^3 = 3^2.5 is much more precise.

If nobody objects, I'm going to remove that one by Ramanujan with fractions and roots, and to re-add the ones about pi, pi^3, pi^4 and pi^5.--Army1987 14:52, 15 May 2006 (UTC)


 * I still don't see how they are "coincidental" rather than simply "unusual". Is there even a distinction to make? For instance, the famous $$e^{i*\pi} + 1 = 0$$ seems almost "coincidental" despite it having a logical proof. I don't know any more. I may leave this page up to you two... --MrHumperdink 23:05, 15 May 2006 (UTC)

Error
Towards the bottom of the page, it is stated that 71^3 = 357911, which is correct. However, this number is referred to as "consecutive primes", which is incorrect. —Preceding unsigned comment added by mstemper (talk • contribs)
 * Good call. I suppose you could say "consecutive odd numbers", but it doesn't seem much of a coincidence. Why not list a factorization of 3579? or 35791113? It's somewhat remarkable that the given number was a perfect cube, but nothing to write home about. Staecker 01:43, 27 May 2006 (UTC)
 * I guess it meant three five seventy-nine eleven, but so, why are they "consecutive"? --Army1987 08:53, 27 May 2006 (UTC)
 * 357911 also consists of three primes (which is amazing since the LHS contains a third power), when written in base 111 (three ones in a row!). Pick any couple of numbers and you can make up some "coincidence" between them if you think hard enough. Entries like this one are pointless. Fredrik Johansson 09:22, 27 May 2006 (UTC)

I've spotted two items which I believe are incorrect. The first one is the second entry on the page, "Exp[pi * sqrt(n)] is integer for many n, notably 163". If I compute Exp[pi * sqrt(163)], I get an extremely large number. Also if I interpret it as the base-pi root of n, I get incorrect results. The second 'error' I've found is in 'Other numeric curiosities', where it is stated that phi = -2 * sin(666), while this is incorrect. In fact, on the page of the golden ratio, it is mentioned that -phi = sin(666) + cos(6 * 6 * 6), which seems to work out. 30 August 2006


 * You are right but there is no error. Exp[pi * sqrt(163)] = e^40.10916999113251975535008362290414... = 262537412640768743.99999999999925... which is both a large number and close to being an integer. Meanwhile sin(666&deg;) = &minus;sin(54&deg;)= &minus;sin(3&pi;/10 radians) = &minus;0.80901699... = &minus;phi/2.  At the same time  cos(6*6*6&deg;) = cos(216&deg;) = &minus;cos(36&deg;) = &minus;sin(54&deg;) so the same value. --Henrygb 14:45, 22 November 2006 (UTC)

Spelling of 40

 * 40 is the only number whose (English name) letters are ordered alphabetically.

This has nothing to do with mathematics - not even marginally - and everything to do with language (but only one particular language). It is in interesting fact to bring out at trivia sessions, but it is certainly not a "coincidence" (mathematical or any other kind), and I don't see how it has any place in this article. JackofOz 01:13, 25 July 2006 (UTC)


 * Agree --Alextalk 14:01, 25 July 2006 (UTC)

Integer Powers of Eleven in Decimal Coincidences Section
It seems to me that the identity 1111111111^2 = 12345678987654321 is just a special case of 1^2 = 1, 11^2 = 121, 111^2 = 12321 and so on until the sequence terminates with the 9-digit sequence of ones listed. I propose that this interesting sequence replace the given text. —The preceding unsigned comment was added by Koalaitis (talk • contribs) 22:31, 31 January 2007 (UTC).
 * You're right, I have removed this one. --KnightMove 13:57, 25 April 2007 (UTC)

"As 12² = 144, 21² = 441."
This is hardly a decimal coincidence, for it this completely expected in all integers squares if not for the limitations of the decimal system.

For example, consider $$(2x+y)^2=4x^2+4xy+y^2$$. Substituting in $$x=10, y=1$$ will automatically give a decimal expansion and no further computations, and thus we can easily switch the values to also switch the digits of the outcome. This is why $$12^2=144$$ and $$21^2 = 441$$. But consider $$(5x+y)^2=25x^2+10xy+y^2$$. Substituting $$x=10, y=1$$ will not be the reverse of $$x=1, y=10$$. Why? Because 25 is not a digit and thus these substitutions will not automatically form a decimal expansion. This so-called "coincidence" that only $$12^2 = 144$$, $$21^2 = 441$$ produce such a result (which is incorrect, as $$13^2 = 169$$, $$31^2 = 961$$ is also similar) is a result of the limitations of the decimal system, not by mere chance. As another example, consider the number base 26. In base 26, 25 is a digit and thus substituting $$x=26, y=1$$ will automatically give a base number expansion in $$(5x+y)^2=25x^2+10xy+y^2$$, and thus $$x=1, y=26$$ will produce a reverse:

$$(15_{26})^2=1AP_{26}$$ and $$(51_{26})^2=PA1_{26}$$

—75.176.165.18 01:41, 24 August 2007 (UTC)

Only powers that differ by n
In addition to the mentioned case of 5^2 + 2 = 3^3, I recall reading that 2^3 and 3^2 are the only non-trivial natural powers with a difference of 1, and 5^3 and 2^7 the only with a difference of 3. I dunno what exactly happens with larger numbers; you can just start picking up power pairs at semi-random (3^3 + 5 = 2^6; 11^2 + 7 = 2^7; 10^3 + 24 = 2^10; 3^5 + 100 = 7^3...) but I haven't found any number cropping up twice, between different powers at least (square differences can account for just about anything non-prime, and cubes a lot too). Not that I've looked very hard. I digress tho; my points being:
 * is anyone able to verify that for n = 1, 3, there are no other solutions either?
 * for that matter, a citation for the case mentioned in the article might be nice too, as it's not exactly a trivial claim.
 * is anyone able to provide examples of multiple cases of some difference between differing powers?
 * or cite a theorem or hypothesis on their non-existence?

BTW, I don't think approximations such as pi^2 = 10 should have no place in here, mostly because it's not even very good an approximation. How about setting a threshold of accuracy before something can count as a "co-incidence"? --Tropylium 11:53, 27 September 2007 (UTC)

2^15 - 7 = 181^2, 2^7 - 7 = 11^2, 2^5 - 7 = 5^2, 2^4-7 = 3^2, and 2^3 - 7 = 1^2, is probably interesting.

In terms of pi^2 = 10, i think historically attested values, like 3 1/8 (Babylon), 3 19/256 (Egypt), and things like 7^7/8^6 (using humble numbers), have a aplace in 'approximates'.

I still think 'approximate' is better than 'coincidence' here, because often the value used is selected out of many available forms, to get an unavaliable number. --Wendy.krieger 08:10, 1 October 2007 (UTC)

Pi Appromimates
7^7/8^6 in octal, 3.110367 vs pi 3.110375524210, is the best appromimation of pi in 'humble' numbers (primes less than 10), in numbers less than 1e8.

256/81 is the ancient Egyptian version of pi, derived from subtracting 1/9 of the diameter, and squaring to get the area.

1521/484 is a very good appromimation to pi in square numbers. A square of edge 44 inches, has the same area as a circle of diameter 39 inches.

20*sqrt(2)/9. An ancient value of pi, which equates a quadrant arc of 10, where the inscribed circle is 9. --Wendy.krieger 07:49, 1 October 2007 (UTC)

Some Curios
6^phi = phi^6 ~ 18. One can start a multiplicitive fibbonacci-series, from 2,3 which passes through 6, 18, 108, 1944 etc.

ln 17 = 17/6. This makes log_17 of e = 0.6000 base 17.

US dry bushel = 14-inch cylinder [cylinder 14 inches high, diameter] (actually 2738 cyl inch, not 2744), is inscribed in a 14 inch cube, representing the heaped bushel (2744 cu in, vs actual 2747.715 in). Also, the wine tun (252 US gallons), is very close to a 42-inch cylinder (exact for pi=22/7) 1 tun = 58212 cyl in, 42 inch cylinder = 58188.6 cu inches.

One hoppus foot = 36 litres, [= 36.054128], 1 kilolitre = 45 cylinder feet (eg cylinder 3 ft diam * 5 ft high), makes pi = 20.sqrt(2)/9. --Wendy.krieger 08:02, 1 October 2007 (UTC)

Joule
That the Joule is an SI unit, is indeed a cooincidence. We know this, because the Joule is part of an older system (Volt-Ohm-Second), where the unit of energy happened to be equal to the then unfashionable "mecahnical" definition of energy, when using the already MKS units.

We see that the Volt-Ohm-second system was used with any measure (eg V/ft, V/cm), the units being an arbitary unit not derived from Length, Mass, Time. Gustav Mie proposed a system V, Ohm, s, cm (mass = dekatonne), and Giorgi suggested V, Ohm, s, m, (mass = kg). This allowed MKS units to be used as a scientific measure, rather than the technological units).

Wendy.krieger 07:19, 5 October 2007 (UTC)

Music: Phi and semitone
Hello Dicklyon,

Once again, you have reverted my contribution. I do not believe this falls within the WP:OR category, since it is based on simple mathematical facts. (It is not a "point of view", but a simple observation, and statement of self-evident fact. I would also like to emphasize that the numbers are not randomly chosen.

In this particular Article, Mathematical coincidence, "references" often rest within the simple mathematics presented, and a clear statement of the "margin of error" involved.

Certainly, if a section of this Article should be reverted, it is the one immediately following mine ("Concerning base 2"), which reiterates common knowledge in a needlessly complex manner. Obviously $$ 2^{4/12}$$ resembles 5/4...it is the very intention of equal temperament that each resultant interval should resemble its expression in simple harmonic ratios.

Before reverting a contribution to Wikipedia, it is a common courtesy to begin a discussion on the Talk Page of the Article, inviting the views of others. I will therefore retract the term "rude", and instead refer to your comments and actions as "hasty".

I am therefore requesting that you avoid another hasty revert, and open the matter to discussion. I am therefore copying this message to the talk page of the Article. Prof.rick (talk) 08:30, 28 December 2007 (UTC)


 * You have ignored my comments on this new-found coincidence of yours before, so here's the latest, copied from your talk page:


 * I have now reverted the link to mathematical coincidences after reverting all the recent WP:OR there. Sorry, it's hard not to be a bit rude when dealing with such blatant violation of policy. Dicklyon (talk) 06:15, 28 December 2007 (UTC)


 * Just to be clear, such coincidences are a dime a dozen. Given two real numbers x and y, find exponents m and n such that $$x^m = y^n$$ by solving $$m \log x = n \log y$$ for integers m and n to whatever degree of precision you want by expanding the continued fraction of $$\frac{m}{n} = \frac{\log y}{\log x}$$ to whatever convergent is as close as you want.  For example, with x being the semitone and y being phi, $$\frac{m}{n} = \frac{\log \varphi}{\log 2^{1/12}} = 8.33090296$$ which has a close convergent at 25/3.  So what?  Works with any other numbers you pick, too.  It's certainly far short of a coincidence. Dicklyon (talk) 06:23, 28 December 2007 (UTC)


 * And yes I agree that there are other sections of this article that should be removed, too. Any coincidence not reported as such in a reliable source should be removed, by standard wikipedia policy.  But maybe we should add citation needed tags first, and give people a chance, since these have mostly not been declared to be newly made up. Dicklyon (talk) 08:38, 28 December 2007 (UTC)

Repeated reverts of "Music: Phi and semitone"
Dickylon,

Is is YOU who has reverted my contribution 3 times! (Are you not subject to the rules?) Despite your refusal to discuss this matter on the "Mathematical coincidence" talk page, I will post a copy of this response there.

When reversing an unexplained (or weakly explained) revert, I do not feel an explanation is necessary to undo the revert. The onus is upon you to discuss the reason for a revert, or, better, to open discussion before reverting.

Please, do not attempt to discuss this matter further on my talk page, but use the talk page at Mathematical coincidence. (That is it's purpose!) Apparently you made comments on the talk page of "Golden Ratio"...let's stick to the talk page of the Article on which my contribution stands/doesn't stand/stands/doesn't stand/stands/doesn't stand, etc.

I suppose I have been naive. I believed that Wikipedians work together for a common cause, and rarely initiated such a conflict. I have gathered that your intention is merely to quash a fellow Wikipedian's contributions. I have made mistakes on Wikipedia in the past, and have readily acknowledged my errors. I am prepared to do so in this instance, if erros should exist, but firmly believe the matter should be open to debate by fellow Wikipedians.

Yours very truly, Prof.rick (talk) 09:59, 28 December 2007 (UTC)


 * Sounds like you need to bone up on how to read histories and talk pages. Anyway, the explanation is above now. Dicklyon (talk) 16:59, 28 December 2007 (UTC)


 * Your contribution needs to be removed again. It is in no sense a mathematical coincidence; 25/3 is just a rational approximation to the ratio of logs of a pair of numbers that fascinate you.  I'll hold off, since as you pointed out I already removed it three times. Dicklyon (talk) 17:03, 28 December 2007 (UTC)


 * Dicklyon, I have tried everything to reach a compromise and amicable working relationship with you, but you remain defiant. In response to your above comments, where did you arrive at the number 25/3?  It is NOWHERE in my contribution. Did you mean $$\Phi^{3/25}$$? Please read more closely before jumping to conclusions! I know how to read histories and talk pages...what is this strange accusation all about, when you post messages on this topic at an entirely different Article (Golden ratio)? Prof.rick (talk) 20:37, 28 December 2007 (UTC)


 * Yes, that's where -- 25/3 and 3/25 are not distinct concepts in the sense that I was referring to. $$\Phi^{3/25}$$ is about a semitone and $$(2^{1/12})^{25/3}$$ is about &phi;.  Coincidence? Anyway, I don't think I've been particularly harsh, non-amicable, or defiant, just trying to get you to understand that wikipedia is not a place to publish original research. Dicklyon (talk) 20:54, 28 December 2007 (UTC)

Hi guys, I just stumbled across this little debate, and I would like to add my comments from a more objective point of view. I've removed the comment on the page which instructs people not to edit it, as I don't think it's appropriate: if people want to improve the section, they should not feel that doing so will be breaking the rules. From looking at the history, it seems you have both broken 3RR... there's a reason this rule is in place: edit wars aren't constructive. I could go find the nearest admin and ask them to block both of you, but I don't think that would be very constructive, and I believe you both acted in good faith. Now for my comment on the actual article content... the whole thing, not just the section in question does need more references, otherwise it will have to be called WP:OR. Instead of deleting it, please try to find sources, or ask the editor where they found the information. It is far better to improve content than remove it. But if there really are no references, the content needs to be removed. Original research is just that, whether it's a simple mathematical observation or otherwise. I'm going to have a bit of a look for references myself, but I know little more than a layman about the topic, so I'm sure both of you, who obviously have an interest in the topic, will be able to make far better contributions. I'll be watching this page for a while, so if you have any responses go ahead and post them here, or on my talk page. Briefplan (talk) 17:23, 28 December 2007 (UTC)

Oh, ok, I forgot to actually remove the comment which I said I had. Thanks Prof.rick. Briefplan (talk) 19:01, 28 December 2007 (UTC)


 * Hello Briefplan,


 * In any case, I removed the comment myself, on your advice.


 * I feel that this specific topic, Mathematical coincidence, is inherently subject to "unwritten" rules of reference. For example, if I were to claim that 0.99999999 approximately equals 1, would a source of reference be required?  It is understood, in Wikipedia, that self-evident facts do NOT require reference. This is certainly the case when "math speaks for itself".  Is a reference necessary to verify that 0.99970836 approximately equals 1? (I have defined the discrepancy.)


 * Considering the very nature of the article (Mathematical coincidence), the observations of editors would seem valid, provided they state the degree of discrepancy between numbers (granted, some express an unreasonable discrepancy).


 * I would disclaim my contribution as "original research", since it simply states a mathematical coincidence (a "nearness of numbers") which anyone might observe.


 * I am open-minded regarding discussion of this matter, but believe my contribution is of value, particularly to musicians (as stated in my "Conclusion"). Let us take our time in resolving this issue, since some Wikipedia content is self-evident, while other material requires reference.


 * Incidentally, no one has broken the 3RR in this case, since 3 reverts are permitted, but must not be exceeded.


 * Thank you for your interest, Prof.rick (talk) 20:17, 28 December 2007 (UTC)


 * Prof.rick has already stated that he just discovered this coincidence and wants to publicize it; what could be more clear WP:OR? I told him the first time I removed it, from Golden ratio, that he could bring it back if he found a source.  He and another guy decided to put it here instead, and have ignored my attempts to discuss it in terms of policy.  I continued to remove it, with edit summaries and talk comments, each time he put it back with no summary and no talk. And I haven't removed it more than three times from this article; I'm hoping someone else will do that. Dicklyon (talk) 19:20, 28 December 2007 (UTC)


 * PLEASE tell me where I, Prof.rick, stated that "I just discovered this coincidence and want to publicize it". If you can quote these exact words from my own writing, I will give you $$/Phi$$ $1,000,000!!!  Honest representation is assumed in Wikipedia, while outright deception is WRONG, whether or not stated in Wikipedia's policies. You realize that this is legally a case of slander? The following statements were made by User:Briefplan prior to the the writing of this paragraph, due to "editorial conflicts". Prof.rick (talk) 21:00, 28 December 2007 (UTC)


 * In this diff you said:
 * I came across the coincidence while examining the Fibonacci series for patterns, spotting the proximity of Phi^3 to 25 ET semitones, and simply wanted to point out the "coincidence". (Phi = about 8.33 semitones.)
 * Is my paraphrase close enough to win me the prize? There's that 25/3 again. Please also review WP:LEGAL before you get yourself in trouble. Dicklyon (talk) 21:10, 28 December 2007 (UTC)


 * Ok, I agree with you. It does look like OR. I'm going to remove it. Prof.rick, the fact that three different people (I'm assuming that what Dickylon says about Golden ratio is true) have objected to the inclusion should be taken as consensus. If you still think it should be included, please rationalise it here in a way which complies with WP policies, rather than just reverting my changes. It's about coming to consensus, so try to convince us why it should stay. Or just find a source for someone authoritative who's come up with the same thing. Briefplan (talk) 19:56, 28 December 2007 (UTC)


 * While you were removing it I was doing a major edit to move some things into a new section on rational approximants. I included a small item on the semitone and phi, not because I think it's not OR or that it belongs, but because as a small item it's not much worse than a lot of the rest of the page contents, and I was in a mood to compromise for now.  I did tag it with citation needed, which is a tag that should also go lots of other places.  By the way, I don't think I said that three different people have objected to it; I haven't looked to see if that's true. Dicklyon (talk) 20:20, 28 December 2007 (UTC)
 * Resigned from project, due to Dicklyon's failure to response to a call for amicable working relations and outright lies.  Prof.rick (talk) 21:53, 28 December 2007 (UTC)


 * The thing you called a lie is documented in bold in your own words, up-thread. Dicklyon (talk) 22:04, 28 December 2007 (UTC)

Cleanup tag
I've tagged both the article and this talk page with cleanup, since there is a ton of unsourced trivia and other stuff that needs to be either sourced or removed. Not every numerical coincidence is noteworthy enough to be in an encyclopedia; only those that have been discussed as such in reliable sources should be here. Dicklyon (talk) 22:13, 28 December 2007 (UTC)

Decimal Coincidences: 1/81
I have corrected 1/81 (previously described as approximating 1.0123456789), because there is no valid mathematical reason to "invent" the "8" (in the 9th decimal place). In fact, 1/81 = 1.01234567901234567901..., apparently a repeating decimal. (Even by rounding, we arrive at 1.012345679, or 1.01234568)

I would have simply removed this one, but feel it is unfair to do so without discussion. Instead, for now I have simply corrected the expression of 1/81. Upon looking at it now, its inclusion in this article seems at least questionable. Prof.rick (talk) 06:20, 14 January 2008 (UTC)

1/81 $$(1/9^2) $$ is correctly expressed as 1.012345679 in the preceding section, "Other mathematical curiosities", so there seems to be no valid reason to allow "about" 1.0123456789 is the "Decimal coincidences" section. Prof.rick (talk) 02:00, 15 January 2008 (UTC)


 * There is little coincidences here. 1/9801 is .00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 ...  This is true in any base.  We see that the '8' disappears because 7 8 9 10 11 ... carries as 7 9 0 1 2 3 ...


 * Of course, if the base is selected large enough, eg powers of (1000/999) gives the pascal pyramid.


 * 1 => 1.001 001 001 001 001 001 001
 * 2 => 1.002 003 004 005 006 007 008
 * 3 => 1.003 006 010 015 021 028 045 <- triangular numbers
 * 4 => 1.004 010 020 035 056 084 120 <- tetrahedral numbers
 * 5 => 1.005 015 035 070 126 210 330 <- pentachoral numbers
 * 6 => 1.006 021 056 126 252 462 793


 * Likewise, 1/(x²-x-1) [eg dec 89] is 0.0112359... for 9989, 0.00 01 01 02 03 05 08 13 21 34 55 90... being the fibannacci series. When x=36, x²-x-1 = 1331, which is a cube.  This is the only recorded example.


 * Likewise, 1/x²-2x-1) eg 997999 for x=1000, gives 0.000 001 002 005 012 029 070 169 408 987 ... This is a series of numbers whose square, doubled, gives one different from another square, eg 169*169*2 = 239*239+1. --Wendy.krieger (talk) 09:07, 8 March 2008 (UTC)


 * You're absolutely right, anything to do with 1/81 are no coincidences. It's just because we work in base 10 and 9 = 10-1 (coincidence?) I've also removed the one about e^i\pi = -1, that's just ridiculous on this page.

Phi and semitone
Dicklyon,

I am wondering why you totally removed my contribution regarding the fact that that 25 ET semitones almost exactly equals Phi^3, without discussion, but then returned this information to the page in your own words.

I would have been open to discussion, and agreeable to stating the coincidence in a concise manner. Prof.rick (talk) 02:51, 15 January 2008 (UTC)


 * Are we a bit forgetful? See the section  above.  My compromise was to allow a brief mention in the new section where it fit, but even that probably doesn't belong, based on WP:NOR. Dicklyon (talk) 03:54, 15 January 2008 (UTC)


 * Are we a bit sarcastic? I don't have to review the facts...you know them. Yes, you "compromised"...a very polite way of describing your actions. I have nothing to add at this point, but await discussion by other editors. Prof.rick (talk) 05:16, 15 January 2008 (UTC)

Discussion: (Phi^3)/4
Dick,

I would like to leave both this and YOUR entry above it open to discussion by other editors. Is this reasonable?

My most recent addition to the article is a "mathematically evident" fact (such as "0.333 approximately equals 1/3"). I have found, in working on other articles with other editors and administrators, self-evident facts do NOT require a published source of reference. The argument is not that we must cite a published source for every single word or figure we enter into Wikipedia, but that the material we present is VERIFIABLE. If you, or anyone, does the very simply math of my recent entry, you will realize that it is unquestionably self-verifying. (This is hardly "original research".)

If we leave these matters open to discussion, I am certain an agreement and/or concensus can be reached, one way or the other. If not, then we can eventually ask for arbitration. (Give the process time!)

With patience, and in continuing dedication to Wikipedia, Prof.rick (talk) 05:09, 15 January 2008 (UTC)


 * Sure, discussion is what we're here for. In the context of "coincidences", I think what we want for "verifiability" is not the checking of the arithmetic, but the verification that something is a "coincidence" in the sense that it is interesting enough to have been remarked on.  If we open the article up to all true mathematical relations that anyone wants to claim is a coincidence, it will be out of control.  Much of the discussion above has been about trying to get back some control over the creeping content, and it seems to me that one obvious filter we want is that we only include content that has a source.  I realize that not everyone shares that view. Dicklyon (talk) 05:24, 15 January 2008 (UTC)

If we are here for discussion, WHY do you delete contributions as soon as they are posted? Again, there was no discussion preceding this recent deletion. How can others have the opportunity to express their views? And what happened to your "compromise"?

The question arises, what is "interesting enough to have been remarked upon" (your criterion for inclusion, according to what you consider of interest)? Certainly, nobody wants to see the article run out of control. However, discussion would lead to the elimination of creeping content. I distinctly feel that you have forced your own concept of "filtering" upon the article. As you said yourself, not everyone shares your view. That is the very reason the material should remain, until discussion by other editors results in agreement, concensus, or even arbitration.

I leave this matter in your hands...to revert the article to the point where both of our statements regarding Phi and the semitone are included for the process I have clearly defined. This is a plea, as a fellow Wikipedian. I'm really unsure what alternatives exist. Prof.rick (talk) 06:00, 15 January 2008 (UTC)


 * In the new item you added the "Phi" was a new name and symbol for the "phi" in the item above it, but meant the same number as opposed to the conjugage golden ratio that it looked like. And there was no explanation of what it was, or why you thought it was a coincidence, or how it fit in that section.  But basically it was another bit of your original research, as far as I can see; if there's a source for it as a coincidence, cite that and it will be OK.  As to discussion, yes, I am often very brief in edit comments when reverting junk, but always willing to talk it over as needed; see WP:BRD.   Dicklyon (talk) 06:54, 15 January 2008 (UTC)

Fourth
I went ahead and took out the item on the fourth, since it follows trivially from the one on the fifth (since a fourth is an octave minus a fifth). And it was just a magnet for elaboration that had nothing to do with mathematical coincidence, as we saw. Dicklyon (talk) 00:59, 16 January 2008 (UTC)

Good. That will save a lot of explanation of the differing detunedness of the tempered fourth and fifth. In fact, the whole section on musical intervals seems unnecessary. (It reads as if someone just discovered the basis of equal temperament for himself.) Equal temperament evolved over centuries of mathematical analysis and experimentation, and is no mere coincidence. Prof.rick (talk) 01:23, 16 January 2008 (UTC)

Simple Numerology
Dicklyon comments, "Simple numerology is not mathematic coincidence." However, if one omits the relation between 8! and the number of minutes in a month, then one would also omit the relation between 10! and the number of seconds in 6 weeks, for the same reason. --Robert Munafo (talk) 01:13, 9 February 2008 (UTC)


 * Yes, I agree. I'll take it out. Dicklyon (talk) 07:19, 9 February 2008 (UTC)

Good -- so, can we take out some of the other stuff for the same reason? The definition given at the top, using the words "lacks direct theoretical explanation", seems to indicate that many of the other items are suspect. Consider for example the one that says 10! equals 6! times 7!. I can give a pretty direct explanation of that (smile) Robert Munafo (talk) 18:49, 9 February 2008 (UTC)


 * Yes, I think so. This article is a magnet for numerology, which detracts from the topic. Dicklyon (talk) 19:07, 9 February 2008 (UTC)


 * How is that one numerology? It's easy to verify but it does not seem trivially apparent to me that there should exist x, y, z so that x! * y! = z!. (Any larger examples? Doesn't look likely because you'll need to include an equal amount of all primes on both sides.) Additionally, there's no symbolism of extra-mathematical significance used in there... I'm not convinced it's necessarily a coincidence but it seems like an interesting factoid nevertheless. --Tropylium (talk) 12:36, 17 June 2008 (UTC)

I kind of felt that way when I first viewed the page. It is a bit junky and disorganized, but even after that is fixed there is a wide range of different types of "coincidence" present. This page discusses the idea: Coincidental Taxonomy by Mark Zimmermann. I think we can characterize these factoids in a similar way. For example (using Zimmerman's nomenclature), $$\pi^2=10$$ is "mere" coincidence, while the nearly-interger nature of $$e^{(\pi\cdot\sqrt{163})}$$ would be "deep". I think both of those belong on the page. The item I added that sparked this conversation is "definitional" or "causal" or perhaps "trivial". Robert Munafo (talk) 01:35, 11 February 2008 (UTC)


 * Some of my stuff is being rejected as original research. Consider the esthetics of 13!=2^10*3^5*5^2*7*11*13=100100*(250+6)*(250-7)=6227020800, and of the related pair A) (365+1/4)^2=3^7*61+9/16 and B) (365+1/4)^4=17797577732+7^2/2^8.Julzes (talk) 02:27, 25 May 2009 (UTC)
 * (10^5+10^2)*(10^3/4+6)*(10^3/4+7) isn't really that big a deal. Given four arbitrary powers of ten and three~four single-digit integers, you can represent a whole lot of numbers really. That (a/d)^2 coincides next to a multiple of a large power of 3 is somewhat interesting, but not all that much; and I'm afraid don't see at all what's special about the last? -- Trɔpʏliʊm • blah 10:00, 25 May 2009 (UTC)
 * I'm sorry, but it is primarily the coincidence of seven 7s and the connection to A (where likely the most symmetrical transformation of the first six digits as a mixed fraction raised to a power (of 2408 possibilities according to my effort)) that makes B appealing. There are other things, but they are not the result of pure inspection.  The fact that the sum of the digits of both numerator and denominator of B are 13 and that numbers congruent to 13 modulo 32 in place of 365 will produce the same fractional part struck me at least.  I am currently doing research on an even stronger coincidence problem related to B that may settle the issue, if an issue exists.  But, as I said, I am just looking at what can be seen of the pairing of A and B by inspection for possible inclusion in the article.  As I see it, the article's introduction could include discussion of it at some later date, before going on to all of the other reasonably neat observations.  I don't judge the first as spectacular as far as coincidences go.  It's a nice way to get from thirteen-factorial to its value--as a semi-mental calculation--with a little additional appeal is all.Julzes (talk) 22:34, 25 May 2009 (UTC)

987654321
For the record, 987654321/123456789 ~ 8 is not a coincidence, it works in any base. —Preceding unsigned comment added by 99.226.91.117 (talk) 03:59, 21 March 2008 (UTC)

Humourous? Really?
"This can be told humorously as: Take the number "1234", transpose the first two digits and the last two digits, so the number becomes "2143". Divide that number by "two-two" (22, so 2143/22 = 97.40909...). Take the two-squaredth root (4th root) of this number. The final outcome is remarkably close to π (within about one part in a billion)." (My bold). In what way is this even slightly humourous? If someone tried to tell me a joke like that at a party I'd have to try remarkably hard not to punch them in the face. El Pollo Diablo (Talk) 13:01, 23 October 2008 (UTC)

Numerical Coincidence
$$10^2 + 11^2 + 12^2 = 13^2 + 14^2$$

References needed to establish notability
I've added tags to all the coincidences listed in this article that do not have a reference to a reliable source to establish notability. This article has had a tag at the top for almost three years now, and it's time to fix it. Please understand that it is not the numerical truth of the statements that must be verified, but their notability. There are infinitely many mathematical "coincidences" that are true, but very few, relatively speaking, have been previously published by reliable sources as notable examples. For instance, yesterday I was playing around and discovered that $$(\pi\cdot e^\pi)^2$$ is within 0.1% of 5280, the number of feet in a mile, which is numerically true, but not notable.

I'm going to come along in a week or two and remove all "coincidences" in this article that have not been provided with a source to establish notability. Consider this fair warning. —Bkell (talk) 22:36, 25 May 2009 (UTC)


 * Please consider this as an ancillary article and Do Not Tamper With Exceptional Effort In A Draconian Manner. If exceptions are not made for some internal attempt at notability, wikipedia will become drier than even you seem to want.


 * I recommend instead that an "Internal Notability" or some other appropriately titled section be placed either at the beginning or the end of the article. Something about a standard of practice in the writing/editing of the article--a disclaimer on notability.  Remember that people will want to see the things on the page as it stands.  I personally won't engage in any edit-war you may start, but I would guess that that is what you will have on your hands.  Remember that wikipedia is not the place to prove a point, and I would tender my opinion that that even includes points on the policies of wikipedia itself, within reason.Julzes (talk) 23:15, 25 May 2009 (UTC)


 * I would also like to advance a not entirely new (please help with source(s)) hypothesis that those who find notable (under broader than wikipedia standards) mathematical coincidences may be channeling. Is God a mathematician?Julzes (talk) 02:12, 26 May 2009 (UTC)


 * In a related vein, I have posted an article request (Fundamental nature of reality) in the philosophy request stream.Julzes (talk) 20:25, 27 May 2009 (UTC)


 * I do want to recognize your concern as valid, as per your 5280 example. Maybe you should consider being a more active editor of this article if such accuracy matters concern you.  Simply taking the position that things should be in print elsewhere before they can be in ANY article in wikipedia seems like bad policy to me.Julzes (talk) 02:21, 26 May 2009 (UTC)


 * You are free to think of it as bad policy, but it's the policy nonetheless. Notability is not the issue (that's an issue for article topics only), but WP:verifiability in a WP:reliable source is the issue.  The only coincidences that belong in the article are those that have been pointed out in reliable sources; otherwise, it's WP:original research.  I support Bkell's plan to clean this up, but I think he shouldn't be in too much of a hurry; many of the items will be found to be sourceable with a bit of searching, but a refimprove tag is never much a spur to editors to do the work; citation needed tags and starting to remove a few is more likely to get some attention, but it will still need some time. Dicklyon (talk) 05:39, 26 May 2009 (UTC)


 * "Notability is not the issue (that's an issue for article topics only)…" Thanks, that's an important point that I overlooked. You're right, of course. —Bkell (talk) 06:32, 26 May 2009 (UTC)'
 * Actually, isn't this all the other way 'round? These being mathematical facts, anyone can verify them by carrying out the relevant calculations. OTOH this shouldn't be simply a page of trivia, and while we can't require notability of facts, we can require someone to have described them as "coincidences". (Ultimately, this works largely to the same effect, so maybe this distinction isn't very relevant.)
 * That said, I do think that at least any deleted parts should be perhaps salvaged here on the talk page, as I'm fairly sure I've seen a decent number mention'd in print at places at some point, but finding said places may be tricky. -- Trɔpʏliʊm • blah 15:59, 26 May 2009 (UTC)


 * This is probably going to entail asking the advice of original contributors, who should have made this effort in the first place. A lot will be obscure (like folded slide rules).Julzes (talk) 17:58, 27 May 2009 (UTC)

I started adding refs; it's a lot of work, but it's what I do. But others need to help. Dicklyon (talk) 07:02, 26 May 2009 (UTC)

I re-edited the introduction to perhaps get the citation request there cleared, and to just improve it. On a different note, it seems to me that many things on the page should remain there even if citations aren't--or have not been found to be within a given time-frame--available. The example that strikes me most is pi^4+pi^5=e^6, but there are others. This article will go through many more edits before it's stabilized, I'm sure. And anybody actually wishing to get the whole history of edits for using this as a partially primary source can always do that, regardless of what policy is.Julzes (talk) 20:12, 27 May 2009 (UTC)


 * That item pi^4+pi^5=e^6 is an example worth discussing. What makes it appropriate for inclusion?  Does the fact than anyone with a calculator can verify mean that it's OK per WP:V?  Certainly not; it says "The threshold for inclusion in Wikipedia is verifiability, not truth—that is, whether readers are able to check that material added to Wikipedia has already been published by a reliable source, not whether we think it is true. Editors should provide a reliable source for quotations and for any material that is challenged or likely to be challenged, or the material may be removed."  This because it's called "verifiability" and one can "verify" with a calculator doesn't mean it suits the policy.  The policy WP:NOR makes this clear, I think.  What really important here is not the near equality, but whether it has been noted as a "coincidence", or at least as an interesting relationship, in a published reliable source.  Otherwise, this article will just continue to collect more such trivia. Dicklyon (talk) 21:10, 27 May 2009 (UTC)


 * Unpublished calculations are permitted by WP:NOR. The article is mostly a list so if an unsourced but mathematically correct entry is questioned then maybe List fact is more fitting than Fact. PrimeHunter (talk) 22:15, 27 May 2009 (UTC)
 * Actually, my experience is that unpublished calculations may be treated as original research if they are used at all to imply something more than the calculations themselves. I'm not continually concerned about that now, though.  I would like this article to be as clean and meaningful as possible, and then possibly later get my interesting results into them (not in a self-serving way).  By the way, we might consider a re-direct of the current article's title to an even broader article on mathematical curiosa/curiosities.  Not every interesting but largely useless mathematical item is a coincidence as we are trying to formualte a definition here.Julzes (talk) 03:11, 28 May 2009 (UTC)


 * "This policy does not forbid routine calculations, such as adding numbers, converting units, or calculating a person's age, provided editors agree that the arithmetic and its application correctly reflect the information published by the sources from which it is derived." In other words, something like the simple percentage error to say how close the coincidence is might be OK, but including a numerical coincidence of a calculation with no reliable source is obviously not going to fall under this policy. Dicklyon (talk) 04:46, 28 May 2009 (UTC)


 * . "Beware of...."  Consider the history of this article by finding MrHumperdink's efforts and community concurrence, and also that a standard is still at issue.  Also, a reliable source is a calculator, under policy.  I am inquiring as to the specifics of the one case I mentioned (originated here above, 7 May, 2006).Julzes (talk) 07:15, 28 May 2009 (UTC)


 * Ah, I didn't know about . Yes, perhaps that is more appropriate. —Bkell (talk) 01:37, 28 May 2009 (UTC)


 * I think "citation needed" is much more clear as an indicator of what is needed. Dicklyon (talk) 04:46, 28 May 2009 (UTC)


 * I think citations are best, but only because of the desire to give proper credit for the originator. As for notability concerns, it seems to me that this article or any descendant as I have recommended should set its own standards and then discuss any exceptions that might be made.  That seems to have been the general practice up to now.  Getting the words surrounding the facts right is, to me, more important than any specific case.  As for the case I cited here and brought to your talk, i just have the reactions, "Wow! Who found that one and how?," and "I am pretty sure it does not mean anything [as my intro indicates], but it's surprising."Julzes (talk) 04:55, 28 May 2009 (UTC)


 * Well, on that note, I'm going to study the history of this article to try to get to the bottom of some of the attribution questions.Julzes (talk) 05:10, 28 May 2009 (UTC)


 * I found where in the stream my pet coincidence came in and by whom, but am waiting on a real source.Julzes (talk) 06:57, 28 May 2009 (UTC)

I am going to do a lengthy rewrite of the introduction that the cognoscenti will recognize as somewhat self-centered, and in it will place a warning about checking other people's as well as one's own calculations; and I will provide an electronic source (specific calculator) for what the article has been CHECKED on (this is the sourceability part, not the notability part). I'm going to tentatively use common sense for standard, in part, and direct readers to WP. How's that? (I won't be done quickly, which is the same as I am waiting for consensus.)Julzes (talk) 18:46, 28 May 2009 (UTC)


 * The article should contain only verifiable content, from published sources; no meta-content, please; keep that on the talk page. Dicklyon (talk) 19:00, 28 May 2009 (UTC)


 * Handled under exception for routine calculations and argument w/r to publication of calculator. "Meta-content" may or may not form policy, depending on the quality of the work, I suppose.Julzes (talk) 19:08, 28 May 2009 (UTC)


 * Routine calculations are not at issue; calculations to establish mathematic coincidences, however, are WP:OR if not supportable by a published source. Dicklyon (talk) 20:10, 28 May 2009 (UTC)


 * We are going to have to disagree on our policy readings, and I did find a published source on coincidences by a CalTech Ph.D., with my pet coincidence highlighted.Julzes (talk) 21:14, 28 May 2009 (UTC)


 * Right now, I would like to chime in again on something old in this talk: "Coincidence" might be better subsumed under "curiosity" with a re-direct.Julzes (talk) 21:17, 28 May 2009 (UTC)


 * I will try to clarify intent: Perhaps mathematics education is the proper angle of the whole article, as per my last line of the current introduction. People could ultimately be guided to knowledge on calculations and their histories by a good writing (of this subsection of a planned "curiosity" article).Julzes (talk) 21:22, 28 May 2009 (UTC)

Simplicity vs. Precision
I want to highlight a particular item recently deleted: sqrt(2)+sqrt(3)~=pi. I think it needs to be in such an article.Julzes (talk) 22:59, 29 May 2009 (UTC)
 * Have you seen it somewhere else? Two decimal places precision in pi is quite low considering the complexity of the expression. Dmcq (talk) 09:41, 31 May 2009 (UTC)
 * No I haven't, and I was surprised to find it was accurate to about THREE significant figures for the first time myself here, since I was one of the best competitive math students in the country in high school. You're not picking on the fact that it technically rounds up to 3.15, are you?Julzes (talk) 11:07, 31 May 2009 (UTC)
 * Your reading of the sum of the first two non-trivial square-roots as complex is a bit mystifying.Julzes (talk) 11:22, 31 May 2009 (UTC)
 * Two decimal places is the number of correct digits after the decimal point. I also find the approximation unimpressive and don't support it without a good source. The article looks like an examplefarm and I think we should work towards only having sourced examples. PrimeHunter (talk) 11:25, 31 May 2009 (UTC)
 * Well, I do tend to agree in general that it's the example-farm you suggest. I personally think it should be part of a larger article and paired down to the best examples with reference to outside for more.  I did not say the case in question was impressive, but I also contest its complexity.  It doesn't get too much simpler.Julzes (talk) 11:34, 31 May 2009 (UTC)
 * 0.15% is the quality of it as an approximation, but it's one that I can't say I see as useful.Julzes (talk) 11:43, 31 May 2009 (UTC)
 * What about simple facts that aren't here? Most everyone you or I know knows about the first ten digits of e, but that doesn't mean it doesn't deserve mention, and the common logarithm of 2 is inaccurately handled in the article.Julzes (talk) 11:47, 31 May 2009 (UTC)
 * An easy estimation of its complexity is given by how many keys you'd have to punch on a simple calculator - 2 root 3 root plus to cut it down using reverse polish. There's 16 keys so that's a 5 digit hexadecimal number whereas the value got had only 3 significant decimal digits. Seems pretty poor to me. I'm sure doing things in binary and using an arithmetic code would make it considerably better but it isn't exactly outstanding. Dmcq (talk) 12:03, 31 May 2009 (UTC)
 * I'm not going to argue with that.Julzes (talk) 12:15, 31 May 2009 (UTC)
 * I'd say my way of looking at it is that the three irrational numbers concerned are apt to be memorized to several digits by many children, yet they might only see the relationship if it's pointed out. It's not critical.Julzes (talk) 12:18, 31 May 2009 (UTC)
 * Has anybody found good textual justification for this article's existence in a single place? There was, in my earlier reading of material that no longer resides in the article, a webpage by a Californis Institute of Technology Ph.D. dealing with the matter.  But if that's about as deep as it goes, it seems almost as though there is no known source justification for the article's existence at all.  No single one, at least.  I certainly am not suggesting this for AfD, but it does almost seem that I and other participants here are making some kind of exception with it.  I have no doubt that this material belongs here in some form, but a recent modification to slightly improve the definition in the lead (by me), which is just a 'common sense' application of what 'mathematical coincidence' means (by including a type in addition to near-integer reals, where a large integer is found to satisfy two relatively simple and not related criteria) should be in any 'real literature' on the subject.  Also, should a mathematics coincidence to a non-mathematical object be considered mathematical?  This is part of the current content, but for something to be coincidental it has to have contact with mathematics to achieve notability. Even if it is a unit-measurement coincidence with no clear reason, it would seem that it is a coincidence but not a mathematical one.  I wish I had a lot of time to devote now to fixing this (these) issue(s) myself, and others, but I do not.  I have made some sort of commitment to wikipedia in other languages that may dovetail, but I don't have any expectations it will.Julzes (talk) 15:28, 18 June 2012 (UTC)

Coincidences in MathWorld
There are a number of the coincidences which are listed in MathWorld, e.g.
 * $$\left(1+\frac{1}{6400}\right) \sum_{k=1}^8 \frac{1}{k} \approx e$$.

in, and the attributions there are to personal communications. Is a reference to the Mathworld articles ok or should the items just be removed and this article just point to the Mathworld articles as see external sites? Dmcq (talk) 09:25, 31 May 2009 (UTC)
 * Any citation is better than none. Let's get what we can done.Julzes (talk) 11:19, 31 May 2009 (UTC)
 * I think would be best to go to the sources cited in Mathworld where they exist. As you point out, some are unsourced technically ("personal communication").Julzes (talk) 12:30, 31 May 2009 (UTC)
 * The very first item in your article choice is an example of a "coincidence" I would add. That's pretty neat.Julzes (talk) 12:33, 31 May 2009 (UTC)
 * It's a "simple" application of (1+1/n)^n-->e. It's sort of sourced.Julzes (talk) 12:41, 31 May 2009 (UTC)

Why is Pi^2 so close to 10?
Maybe this paper by Noam Elkies can be added. http://www.math.harvard.edu/~elkies/Misc/pi10.pdf —Preceding unsigned comment added by 84.127.78.170 (talk) 23:48, 9 November 2009 (UTC)
 * Excellent – thanks, done!
 * —Nils von Barth (nbarth) (talk) 03:57, 1 December 2009 (UTC)
 * Yeah, but what that really means is that it isn't a coincidence after all, so it shouldn't be on this list, no? — Preceding unsigned comment added by 92.67.227.181 (talk) 00:03, 8 June 2022 (UTC)

Re-writing and re-organizing and other things needed
I'm not spending a lot of time on this today, but this article needs a major prose infusion. I would put a few paragraphs together for the introduction, do something about distinguishing between mathematically meaningful coincidences and those which are quite apparently not of that nature, and say something about Pythagoreanism and about distinguishing between data-mining quasi-results and everything else. I would suggest, or will write myself, an introduction that includes a paragraph combining the second through fifth coincidences currently listed. Nowhere does the fact that e=2.718281828... appear, and this would be highlighted in what needs to be said to get this article right. The decision to attack things that smack of numerology in the history of this article is certainly somewhat incorrect. There were once, before I came here, a whole host of supposed coincidences in units. I have not analyzed that material, but I have reasonably good grounds to take another look at it. I'll be back to spend more time on this article periodically (It's not top priority for me).Julzes (talk) 23:43, 3 January 2010 (UTC)


 * I though the main reason things had been removed was that no reliable sources could be found for the removed stuff. It had become a list of peoples own calculations. Dmcq (talk) 00:14, 4 January 2010 (UTC)

I'm not going to quibble with that (I have not researched it yet), and I don't really object to most of the removals. The real problem with the article as I see it is that it may as well have no introduction at all the way things currently stand. A good introduction could shape the way the lists under it are presented. OR, unsourced, doesn't belong here, and anyone who wants this or any other article in wikipedia to include her or his (supposedly) good new research included has to go somewhere else first at least. On the other hand, I would wonder in mystification if an article like this would ignore the decimal representation of $$e \,  $$ for long if all mathematicians did was grow up with a little "Wow!" and then never a single one of them write down the fact that it had that impact.Julzes (talk) 00:48, 4 January 2010 (UTC)


 * A good leader is always an excellent advantage for an article. As to the expansion of e I'm sure there probably is some citation for that if one can find it. In most maths articles one can get away with sticking something fairly obvious like that in but I think this one has to be careful with strong entrance criteria or it'll just degenerate into rubbish. Dmcq (talk) 14:44, 4 January 2010 (UTC)

I won't be doing anything directly for ten days. I'll take any suggestions.Julzes (talk) 19:08, 9 January 2010 (UTC)

I've decided after all not to do any work on this or much else here at wikipedia (unless I see something blatantly false and easy to correct while reading) for a while, so if someone wants to take over they can feel free. The article requires a lot of work, and I don't see myself doing that for a while.Julzes (talk) 15:17, 15 January 2010 (UTC)

One thing at least I see as somewhat off in this article is in the very definition. There are certainly other sorts of mathematical coincidence than the near-integer real type. Examples: A) 3360633 appears as both the first number that is a 7-digit palindrome in 3 bases (also in bases 9 and 11) and as the 8th palindrome amongst sums of composite numbers up to a certain point, where also 33633 is the 6th such number; B) concatenating the positions of the letters in ABRACADABRA yields two 13-digit emirps--4 primes in total by the processes of reversal of letters and of digits; C) the 4th and 44th prime numbers to yield primes twice in succession by inter-base digital translation from both bases 2 and 3 to base 10 are the first to also do so once and twice from base 4 to base 10, and these two numbers also lead with digits 234 (in base 10); and D) the first and possibly only k for which there exist concatenations of the first k primes that are products of k distinct primes is k=10, with 10 numbers as examples (11 distinct concatenations, one pair being equal numbers by virtue of there being two copies of 23). These are just a few I can recall offhand (and all are actually my own research (with light or no sourcing, so I am not advocating inclusion yet), but I am not remembering others), but many exist. I would advocate for nearly as large a definition as conceivable, including making space for a little discussion of 'the surprising usefulness of mathematics'. Has anybody looked at what is being done with this subject in other-language wikipedias? I have not had much time, and it just occurred to me anyway (plus I am not the most polyglot person around).Julzes (talk) 02:34, 20 March 2012 (UTC)

Prime curios
A recent diff stuck in stuff I would never have imagined would have made it into a book I was surprised when told they were in the book "Prime Curios!: The Dictionary of Prime Number Trivia". Could someone who has access to the book verify that and say if it really is full of that sort of stuff thanks as I'm finding it hard to believe anyone would pay good money for that sort of thing.

The main point though is shouldn't there be some better selection criteria if there are books full of trivia like that? The book is entitled trivia and I really don't see that Wikipedia should duplicate its contents. Dmcq (talk) 15:35, 14 June 2012 (UTC)


 * Both those were contributed to the Prime Curios website by 'Merickel' as far as I can make out and the contributor index for the book can be looked at from and it doesn't have Merickel listed. They may have a different name there or some mistake but yes I really would like someone to check now thnks. Dmcq (talk) 15:44, 14 June 2012 (UTC)


 * This article is still a mess. I apologize for raising your hopes, if I did.Julzes (talk) 22:28, 25 June 2012 (UTC)
 * The book is older than Merickel's first website contribution.Julzes (talk) 18:59, 29 January 2013 (UTC)

Assess
Re-assessed to C-class.-- Gilderien Chat&#124;List of good deeds 14:08, 14 July 2012 (UTC)

FAILED TO PARSE
All over the page! HotdogPi 03:13, 8 September 2012 (UTC)

a couple of physics coincidences

 * To convert meters/second to miles/hour, multiply by $$\sqrt{5}$$ : correct to 0.04%
 * Doubling a power quantity is an increase of 3 dB; doubling a  field quantity is an increase of 6 dB :  both correct to 0.3%

M-1 (talk) 00:25, 5 October 2012 (UTC)
 * Corresponds to the mathematical coincidence $$\log_{10} 2 \approx 0.3$$.


 * The time required to accelerate from rest to the speed of light (c0) at an acceleration of one Earth gravity (g0) is very close to one year (it's 353.8 days, or 3.1% short of 1 year). --Heron (talk) 09:10, 29 November 2012 (UTC) P.S. I know that this is a naive calculation and ignores the physical impossibility of reaching c0.
 * If it's naive and impossible, why should it be mentioned? Logically, based on a false premise, you can make any conclusion, but that doesn't make it relevant to this article. CodeCat (talk) 14:20, 29 November 2012 (UTC)
 * It's still a scale-factor, causing 1 g acceleration corresponding to a real velocity of $$v \approx \tanh t$$, where v is measured in units of the speed of light and t is measured in years. — Arthur Rubin  (talk) 15:00, 29 November 2012 (UTC)
 * That may be true, but it doesn't have any real-world significance whereas all the others do. It's a mathematical coincidence based on incorrect laws of physics. I mean, I could argue that with a speed of 1000 km/s, it takes a planet almost exactly 1 year to come to a stop, disregarding Newton's first law. Or I could argue that it takes almost exactly 1 million years for the sun's light to turn purple. That doesn't make it relevant to this article. CodeCat (talk) 17:48, 29 November 2012 (UTC)
 * The speed of light issue is further discussed (and solved) below in section Gap9551 (talk) 18:47, 6 April 2016 (UTC)

Example needing new category (?)
I don't see a good place in this article for the fact that 13^2 seconds short of midnight on a 24-hour clock is the time 23:57:11 (which, if the standard has not changed and I did not misread it, certainly satisfies the elementary-computation exception to WP:OR). To the extent it is recognizable as one at all, it's not a heavy-duty coincidence, but it would seem to be a good lead to others of a kind.Julzes (talk) 16:42, 25 June 2013 (UTC)
 * I don't see a coincidence, there. Could you explain?

"A famous maths professor was giving a lecture during which he said 'it is obvious that...' and then he paused at length in thought, and then excused himself from the lecture temporarily. Upon his return some fifteen minutes later he said 'Yes, it is obvious that....' and continued the lecture."
 * — Arthur Rubin (talk) 18:49, 25 June 2013 (UTC)
 * I guess it's about the first primes 2, 3, 5, 7, 11, 13. At 23:57:11 there are 132 seconds to midnight. It's a routine calculation per WP:CALC but we would need a source to claim it's a mathematical coincidence. It would have been nicer without the exponent 2. PrimeHunter (talk) 00:37, 26 June 2013 (UTC)
 * Thanks for answering for me, PrimeHunter. Not that important to me, Professor.  I just got a coincidental sentence in my numerological research saying it was the most likely to make it through (because of other obstacles discussed elsewhere on the quality of sources); so I thought to put it in and then saw there was no particular place for it (and so this talk section).Julzes (talk) 21:02, 26 June 2013 (UTC)

Earth's circumference
According to a recent edit: "The circumference of the earth is equal to forty thousand kilometers, to within 0.2%."

This is correct but it is not a coincidence. The kilometre was originally defined to be 1/10,000 of the distance from the equator to the pole -- see History_of_the_metric_system. Nick Levine (talk) 13:19, 8 November 2013 (UTC)

cos e + sin e
cos e + sin e = about -0.5? Is that an example? — Preceding unsigned comment added by JDiala (talk • contribs) 07:44, 30 January 2014 (UTC)

On the face of it, I'd say so. Correct to within 0.2% M-1 (talk) 23:56, 22 February 2014 (UTC)

Eighth root of ten
The eighth root of ten is very close to four-thirds. — Preceding unsigned comment added by 72.77.58.127 (talk) 18:57, 29 April 2014 (UTC)

Speed of light
The section "speed of light" says


 * Another coincidence is that one lunar year (354 days) of acceleration with 1g gives speed of light: 9.8×354×24×3600=299,738,880.

That doesn't make any sense. You could accelerate forever at 1g and not reach the speed of light. Loraof (talk) 14:44, 9 October 2015 (UTC)
 * True, I made a small note to point that out. Maybe we should remove the whole sentence but let's wait for more input. Gap9551 (talk) 16:32, 9 October 2015 (UTC)
 * I removed it now, because further down in the subsection 'Gravitational acceleration', it is mentioned that g is close to 1 lightyear/year^2, which is physically fully correct and captures the same coincidence. Gap9551 (talk) 18:46, 6 April 2016 (UTC)

366 days squared
The square of the number of seconds in a leap year is close to a power of 10: $$(366 \times 86400)^2 \approx 10^{15}$$ (±0.002%). --MizardX (talk) 15:39, 22 June 2016 (UTC)

21 42 84 - interesting geometrical sequence
2 + 4 + 8 = 14; 1 + 2 + 4 = 7; 21 + 42 + 84 = 147;

Let's swap digits in numbers: 21, 42, 84 to make 12, 24, 48 - another geometrical sequence. Then subtract "second" numbers from first ones: 21 - 12 = 9; 42 - 24 = 18; 84 - 48 = 36;

9, 18, 36 - it is also a geometrical sequence.

Subtracting "third" numbers from second ones: 12 - 9 = 3; 24 - 18 = 6; 48 - 36 = 12;

"Fourth" numbers (3, 6, 12) also form a geometrical sequence.

Subtract "fourth" numbers from "third" ones: 9 - 3 = 6; 18 - 6 = 12; 36 - 12 = 24;

"Fifth" numbers also form a geometrical sequence: (6, 12, 24).

Subtract "fifth" numbers from "fourth" ones: 3 - 6 = -3 6 - 12 = -6 12 - 24 - -12

"Sixth" numbers also form a geometrical sequence: ((-3), (-6), (-12)).

Subtract "sixth" numbers from "fifth" ones: 6 - (-3) = 9; 12 - (-6) = 18; 24 - (-12) = 36;

"Seventh" numbers also form a geometrical sequence: (9, 18, 36).

We can make that iteration any number of times and we would always receive a geometrical sequence. — Preceding unsigned comment added by Rabbitsquirrelcat (talk • contribs) 18:28, 14 September 2016 (UTC)

Text too
3*1013*1669*211317915670188235207471*917594864466917047064519*349052954223539065525171338860405905128439 = 10000*(1092738277*314158989541307472007949836495783819190879794835648629101616910339200019752591795513)+101 (something about 'C-O-N-G' as a transliteration). OR, so not. — Preceding unsigned comment added by 96.83.240.59 (talk) 13:16, 22 February 2017 (UTC)
 * Note: 84-digit, two 24-digit primes, leading digits of pi, 3 years after Hanoi's being named capital of Vietnam, Room 101 in 1984 by G. Orwell who's understood to have chosen title by inverting final pair. So, a mix.  Notable for page, highly NO! Inexplicable historical baggage.Julzes (talk) 20:43, 22 February 2017 (UTC)
 * Result out 13 hours and 16 minutes ago.Julzes (talk) 21:43, 22 February 2017 (UTC)

Conway's climb to a prime
The prime factorization of 13532385396179 is equal to 13×532×3853×96179 which uses the same digits and ordering as the number itself. I don't know if this is considered a coincidence or not. — Preceding unsigned comment added by 71.179.19.89 (talk) 23:03, 14 December 2017 (UTC)


 * Indeed; added to the Decimal coincidences section. Renerpho (talk) 15:27, 15 April 2024 (UTC)

Power towers?
2^2^3^2^-1 = 2^2^√3 = 10.000478217... ~= 10. e^11^3 begins with four 1s 7^3^6 and 2^2^11 are close in magnitude? 6^2^7 almost a googol? 2^2^666 ~= 10^10^200 (actual value 10^10^199.96458...), involves number of the beast 71.179.19.89 (talk) 23:27, 14 December 2017 (UTC)

Square root of 62 inches to centimeters
√(62) inches ≈ 20 centimeters, with an error of 0.0001% (0.2 microns). A more accurate value is 19.999979999989999989999987499982... centimeters. Note that most of the digits in the first 32 digits are nines. 71.179.21.46 (talk) 19:33, 27 August 2018 (UTC)
 * ✅ Gap9551 (talk) 19:38, 6 September 2018 (UTC)

100 decimal = 64 hexadecimal, 100 octal = 64 decimal.
Something useless I noticed one day, but is strangely interesting. --Skintigh (talk) 02:49, 17 January 2019 (UTC)

Coincidences regarding Mersenne primes
The 13th Mersenne prime is $$2^{521}-1$$, and $$\sqrt[13]{521}\approx\frac{1+\sqrt{5}}{2}$$. In addition, 521 is the 13th Lucas number.

The last nine digits of the 43rd Mersenne prime, $$2^{30402457}-1$$, contain each digit from 1 to 9 exactly once: 652943871.

—Jencie Nasino (talk) 02:10, 22 July 2019 (UTC)

why 163?
perhaps this is a dumb question, but why is 163 a category? is it very fundamental or something, or do i just not get some reference Ajlee2006 (talk) 13:20, 23 August 2020 (UTC)


 * No, not really. I've removed two of the three examples under it and combined the last entry into the previous subsection. –Deacon Vorbis (carbon &bull; videos) 13:31, 23 August 2020 (UTC)

Another coincidence

 * $$\sqrt[\pi]{4}\approx e^{W(\ln2)}$$. Correct to about .317%.
 * The error's error is about 1.024*π&#8240;.Alfa-ketosav (talk) 20:10, 22 September 2020 (UTC) Edited by Alfa-ketosav (talk) 08:35, 18 February 2021 (UTC) $$\because$$ it said the word pi instead of the letter pi.
 * Please don't list random stuff like this here; without a source to demonstrate any noteworthiness, it can't be added to the article. Otherwise, there's no limit to what we could add.  See also WP:NOR. –Deacon Vorbis (carbon &bull; videos) 23:53, 22 September 2020 (UTC)

Nonce word suggestion
I am here to propose that 'decimaleasemeta' be used, eventually, as synonymous and eventually clearer than what is meant by 'mathematical coincidences' collectively here. It is difficult to come up with a mathematical coincidence that is known to toddlers, for example, before the expression '2.718281828' -- the 10-digit calculator's unhidden value of exp(1) -- is seen. I have no use in mind; this would be new. It just forms an abbrevatory single word out of 'decimal e's meta', 'meta' not really being a proper word generally other than in informal ways.

Something which I nicknamed "Gatomon sequence" (it is associated with additive prime numbers)
Gatomon sequence (aka Tailmon sequence) - ascending or descending arithmetic progression or geometrical sequence formed by additive prime numbers which all have the same sum of digits which is also an additive prime number.

5, 23, 41 and 191, 227, 263 are (examples of) Gatomon sequences (Tailmon sequences). I may think that suggested names are "cool" and "funny" because they commemorate |a cute catlike creature from Digimon franchise by naming such sequences as "Gatomon sequences" ("Tailmon sequences"). It is rather not surprising that I do not know if there are any other Gatomon sequences, even in decimal system.

5 = 5; 2+3 = 5; 4+1 = 5. 5 - an additive prime number. 5, 23, 41 - common difference is 18.

1+9+1 = 11; 2+2+7 = 11; 2+6+3 = 11. 11 - an additive prime number (1+1 = 2). 191, 227, 263 - common difference is 36.

--Rabbitsquirrelcat (talk) 20:08, 18 December 2021 (UTC)

Nanocentury
(I'm putting this in a new section because the comment I'm replying to is years old and my reply will be a leaf in the forest if posted inline.)

>$$\pi$$ seconds is a nanocentury (ie $$10^{-7}$$ years); correct to within about 0.5%

>* ''Yes, or 3, or 3.1, or... basically, nothing special about pi here.''

You conveniently forget to mention that although a nanocentury is 1.004π s, it's 1.018·3.1 s and 1.052·3 s.

Insofar as it's a coincidence there is in some sense nothing special about π here, but that's basically just an argument for deleting this entire article. But if you compare it to π, it's remarkable how much closer it is than compared to even 3.1 and that's kind of the point of a coincidence. So I don't understand why it was removed.

Explaining e^π − π ≈ 20
The section concerning π and e mentions that $$e^{\pi}-\pi\approx20,$$ without giving an explanation. I would propose the following addition, once the concerns mentioned below are addressed:

This is explained by the fact that $$\sum_{k=1}^{\infty }\left ( 8\pi k^{2}-2 \right )e^{\left (-\pi k^{2} \right )}=1,$$ a consequence of the Jacobian theta functional identity. The first term of the infinite sum is by far the largest, which gives the approximation $$\left (8\pi-2 \right )e^{-\pi}\approx 1,$$ or $$e^{\pi}\approx 8\pi-2.$$ Using the estimate $$\pi \approx 22/7$$ then gives $$e^{\pi}\approx \pi+(7\cdot\frac{22}{7}-2) = \pi+20.$$

That's fairly straight forward. I am not adding it though, for two reasons. First, I am concerned about WP:OR and WP:VERIFY. I don't have a usable source; in fact, the source we use claims that This curious near-identity was apparently noticed almost simultaneously around 1988 by N. J. A. Sloane, J. H. Conway, and S. Plouffe, but no satisfying explanation as to "why" e^π-π≈20 is true has yet been discovered. And second, it is not my idea. I've first seen it in a YouTube comment. Renerpho (talk) 19:38, 9 September 2023 (UTC)


 * As a side note, a proper source for the statement that no explanation is known, which doesn't come with a reference itself on, would be Maze&Minder, 2005, page 1. Renerpho (talk) 05:37, 26 November 2023 (UTC)
 * The source has been updated accordingly yesterday. Given there were no objections, I'll add the paragraph as initially suggested (with some minor adjustments to integrate it into the existing list), and tag the section of the Almost integer article that uses this as an example of a mathematical coincidence (which this is, in fact, NOT) as needing to be updated. Renerpho (talk) 12:23, 30 November 2023 (UTC)
 * There is still some element of "mathematical coincidence", because the approximation is an order of magnitude more precise than would be expected. That $$e^{\pi}-\pi$$ is almost an integer is not a coincidence though. Renerpho (talk) 12:59, 30 November 2023 (UTC)
 * The relevant part of the "personal communication" can be found here (including a derivation): https://pastebin.com/VzqPG5Gk Renerpho (talk) 14:19, 30 November 2023 (UTC)
 * You should not be taking any credit for this. It is not "using an idea" that I gave; it is copying my derivation and calling it your own. Adomanmath (talk) 21:44, 1 December 2023 (UTC)
 * Who called it their own? Where did they do that? JBW (talk) 21:55, 1 December 2023 (UTC)
 * The attribution on the "π and e" page says "D. Bamberger, pers. comm., Nov. 26, 2023, using an idea from A. Doman". That implies that he contributed to the proof of the identity, which he did not. I explained exactly how to prove it and did write a full proof months before he did so. Adomanmath (talk) 21:59, 1 December 2023 (UTC)
 * What "π and e" page? Where? Are you referring to a Wikipedia article? If so what is its title? If not, what has it to do with anyone here "calling it [their] own"? JBW (talk) 22:27, 1 December 2023 (UTC)
 * Yes, it's https://en.wikipedia.org/wiki/Mathematical_coincidence#Containing_both_%CF%80_and_e. Adomanmath (talk) 22:29, 1 December 2023 (UTC)
 * (1) What has that to do with anyone here taking credit for anything? (2) Where is a reliable source attributing it to you (whoever you are)? JBW (talk) 22:38, 1 December 2023 (UTC)
 * I have taken credit for exactly one thing, and that is contacting Eric Weisstein (of MathWorld) with a fleshed out version of the idea you presented in your YouTube comment. To quote my email to him, in which I respond in part to the question of attribution that Eric had brought up in a previous email, and which I have already shared above (see the pastebin link):
 * Credit for the idea to differentiate the Jacobian identity at τ=i goes to Aaron Doman, a.k.a. @MathFromAlphaToOmega, who mentioned it in a comment on a video by popular math YouTuber @Mathologer (Burkard Polster). You may want to contact Aaron to ask if they are fine with being credited for it ([contact details removed]). I am okay with being named if you wish to, but all I did was notice that it contradicted what's said on MathWorld, flesh out the details to confirm that it's correct, and of course get in touch with you.
 * The contact details that I removed from the pastebin link are here (your website). Until now, I had assumed that Eric contacted you, to check what kind of attribution you'd prefer. I am sorry if that didn't happen. You can see in that pastebin link what exactly I had told Eric. The rest was up to him (well, I had thought it was up to you, because I told him to get in touch with you). It is sad to see you react so angrily to this.
 * What "π and e" page? I assume that Aaron means the MathWorld article. Where is a reliable source attributing it to you (whoever you are)? I don't know what you're asking for. They are Aaron Doman, and the reliable source is the MathWorld article. The link to the original YouTube comment is already included above. Renerpho (talk) 10:22, 2 December 2023 (UTC)
 * @Aaron, I just sent you an email with further details. Renerpho (talk) 10:30, 2 December 2023 (UTC)
 * For completeness, Aaron's original YouTube comments, posted two months ago, were, quote:
 * There's actually a sort-of-explanation for why e^π is roughly π+20. If you take the sum of (8πk^2-2)e^(-πk^2), it ends up being exactly 1 (using some Jacobi theta function identities). The first term is by far the largest, so that gives (8π-2)e^(-π)≈1, or e^π≈8π-2. Then using the estimate π≈22/7, we get e^π≈π+(7π-2)≈π+20.
 * and
 * I wouldn't be surprised if it was already published somewhere, but I haven't been able to find it anywhere. I was working on some problems involving modular forms and I tried differentiating the theta function identity θ(-1/τ)=√(τ/i)*θ(τ). That gave a similar identity for the power series Σk^2 e^(πik^2τ). It turned out that setting τ=i allowed one to find the exact value of that sum.
 * Eric had asked me to provide more details so he could follow the proof, so I fleshed it out (see the pastebin link). I never claimed the proof was mine. On the contrary, I did everything I could to make sure attribution for it goes to Aaron (including finding out who had made the original YouTube comment in the first place). Renerpho (talk) 10:45, 2 December 2023 (UTC)
 * I may add that I only contacted Eric last week, two months after the original comment, when another user complained on YouTube that we don't have a reliable source to add this to Wikipedia. Renerpho (talk) 11:13, 2 December 2023 (UTC)
 * When you say that you did write a full proof months before he did so, are you talking about a proof you had published? If so then I was not aware of that, and that document would indeed need to be specifically attributed. Renerpho (talk) 11:18, 2 December 2023 (UTC)
 * Eric has adjusted the attribution, it now says: A. Doman, Sep. 18, 2023, communicated by D. Bamberger, Nov. 26, 2023. I hope you can live with this version. Renerpho (talk) 11:56, 3 December 2023 (UTC)

Another coincidence?
987654321 / 123456789 ≈ 8.0000000729 ≈ 8 180.244.139.139 (talk) 16:12, 10 March 2024 (UTC)
 * That could be done in any non-tiny base and follows from how bases work so it's not a coincidence. In base 10: 123456789×8 = 123456789×(10-1-1) = 1234567890 - 123456789 - 123456789 = 1111111101 - 123456789 = 987654312. PrimeHunter (talk) 22:52, 10 March 2024 (UTC)

Coincidence about 7^q = 10^p
Quote from article: << $$\,7^{510}\approx10^{431}$$. This is equivalent to $${\log7}\approx\frac{431}{510}$$ >>

What is the significance of this fact? Of course you can approximate $$\lg 7$$ by infinitely many rational numbers p/q which will lead to approximations $$7^q \sim 10^p$$. Any pair of non-zero non-unit natural numbers (and not only) have the same approximate equialities. One can agree that $$2^{10}\sim 10^3$$ is interesting because exponents are small and precision is quite impressive, but 510 and 431 are large enough to be interesting. — Preceding unsigned comment added by 46.39.51.110 (talk) 11:37, 13 April 2024 (UTC)
 * The same question for << $$6^9 = 10077696$$, which is close to $$10^7 = 10000000$$. Also, $$6^9 - 10(6^5) = 9999936$$, which is even closer to $$10^7 = 10000000$$ >>and << $$\,7^6 = 117649\approx\frac{10^7}{85} = 117647.0588$$ >>.

Removed, per the above, and because no citations were given that would demonstrate how this was notable. Renerpho (talk) 15:13, 15 April 2024 (UTC)

Earth solar orbit
(See section "Planet Earth", added recently about pi*10^7 ) Again, there is no significance in that, no specific of pi there, just about "3" or "3.15", etc. See also discussions above in the current talk page (sections "Not-Coincidences Removed" and "nanocentury"). Please, remove it. 46.39.51.121 (talk) 14:24, 24 April 2024 (UTC)