Talk:Formulas for generating Pythagorean triples

Sum of odd numbers
I don't know if this is a formula or a method:

The sum of the first n odd numbers is n2. If the last odd number of the sum is a square, we have pythagorean triple:

For example:

(1+3+5+7)+9 = 1+3+5+7+9 or 16+9 = 25 190.30.177.192 (talk) 02:45, 18 June 2009 (UTC)
 * :This is the method of Leonardo of Pisa (aka Fibonacci) See Formulas for generating Pythagorean triples III.

Variation on IV. and V.
I wasn't aware of IV. and V. when I developed this.

C2 = A2 + B2

So let A be odd and B be even, given that C must be odd [If C was even, then either A & B are both even, and thus can all be divided by 2, hence not primitive, or A & B are both odd, but the square of an odd (Mod 4) is 1 [(2n-1)2 = 4(n2-n)+1], so the sum of the Mods of the odd squares is 2, which doesn't coincide with the even square (Mod 4) equalling 0.]

Let A = C-x and B = C-y and introduce D, where A = D+y, B = D+x and C = D+x+y. (x is even whilst y is odd, but we will get more specific shortly.)

C2 = A2 + B2 (D+x+y)2 = (D+y)2 + (D+x)2 D = sqrt(2xy) so, A = sqrt(2xy)+y, B = sqrt(2xy)+x and C = sqrt(2xy)+x+y.

Taking a few simple examples before generalizing: (3, 4, 5): x=5-3=2, y=5-4=1, D=sqrt(2*2*1)=2 (=A+B-C) (5, 12, 13): x=8, y=1, D=4 (15, 8, 17): x=2, y=9, D=6 (21, 20, 29): x=8, y=9, D=12

Triads of the form (2n-1, ((2n-1)2-1)/2, ((2n-1)2-1)/2) commencing with (3,4,5) yield y = 1 and x = 2, 8, 18, 32, ...2n2... for n>=1, whilst triads of the form (4n2-1, 4n, 4n2-1) commencing with (3,4,5) yield x = 2 and y = 1, 9, 25, 49, ...(2n-1)2... for n>=1.

Examination shows that triads where HCF(x,y)>1 are non-primitive. —Preceding unsigned comment added by 210.10.131.45 (talk) 04:22, 26 August 2010 (UTC)

Cleanup
This article is in dire need of cleanup. First and foremost, the sections should be given proper titles and not numbers. Some of the sections could also do with some wikification (e.g., converting to maths markup instead of ASCII). I think the sections could also do with some improvements to clarity. 68.76.147.65 (talk) 02:27, 16 October 2010 (UTC)

Sections XIII and V - proposal for reinstatement
Professor Dickson, on Page 169 of his “History of the Theory of Numbers” Vol.II. Diophantine Analysis, Carnegie Institution of Washington, Publication No. 256, 12+803pp read online at University of Toronto here makes only one comment, that his solution is “equivalent to (1)”. (1) appears at the foot of Page 165 as the standard two squares method, which is universally recognised as producing only primitive triples. Dickson does not refer to non-primitives. Professor Loomis in his book “The Pythagorean Proposition” (Pages 19 and 21) (Loomis, E. S. The Pythagorean Proposition: Its Demonstrations Analyzed and Classified and Bibliography of Sources for Data of the Four Kinds of "Proofs," 2nd ed. Reston, VA: National Council of Teachers of Mathematics, 1968), and Professor Paulo Ribenboim in “Fermat’s Last Theorem for Amateurs” (Pages 7 and 8) - (New York: Springer-Verlag, 1999) discuss Dickson’s solution and refer only to primitives and not to non-primitives, while the latter says “Hence necessarily both u and v′ (providing Dickson’s s and t) are squares” thereby saying necessarily only primitives can arise.
 * “equivalent to (1)” refers to Euclid's equation, and to the chapter heading "Methods of Solving $${{x}^{2}}+{{y}^{2}}={{z}^{2}}$$ in Integers", so it appears you have either misread or misunderstood Dickson. Non-primitive triples are (by definition) solutions to both equations.

So all triples were not seen by Dickson and other number theory authors to be calculable by his simple equations. In mathematics, examples are usually considered to be insufficient to prove an important theorem about the production of all non-primitives. (Otherwise Fermat’s Last Theorem would easily be “provable”!). It is probable the Professors may have known that stating the universal generality of non-primitive production would need a proof. A publication to inform readers would surely need to refer to one, either from published literature, or be provided. Otherwise it is mere speculation. A proof was provided in the deleted Entry XIII, and is summarised in www.calculatingpythagoreantriples.org.uk As Dickson did not include non-primitives in his results, the authorship or source of so fundamental a theorem should be provided. The above website was first published in the year 2000 and provides an original source.

It appears that the combination of Entries V and XIII under V is unsatisfactory They have not been fully combined, or give essential information about proof, source or correct mathematical context, and further changes are necessary in the near future. Hoarwithy (talk) 11:29, 22 August 2011 (UTC)
 * Dickson’s method clearly allows for the non-primitives as the example in VI shows.  In fact, when s and t have a common factor, his equations produce ONLY the non-primitives.  Since Dickson’s method also produces all the primitives, we have only to use integer k as a multiple to obtain any desired non-primitive [ak,bk,ck]. Alternatively, we can apply k to st.

However
There are a number of simple, factual, mathematical errors in the statement “Non-primitive triples are (by definition) solutions to both equations”, and the comments made above are based on these errors. The equation (1) Dickson refers to is - $$x = 2mn$$, $$y = m^2 - n^2$$, $$z = m^2 + n^2$$ which, together with its numerous derived methods for generating triples, will only produce a relatively few non-primitive triples when $$x y z$$ are multiplied by a square or half-square integer value. (38 possible non-primitives from the first 500 multiplying numbers, and reducing. A simple test is to try to find the triples 9.12.15, 15.20.25.etc.)  The few calculable non-primitives can be given as examples, but these methods cannot provide more solutions, “by definition” or otherwise. This basic mathematical misconception well proves that examples are not proofs, and is another reason why number theorists never use them when this may infer an un-proved and un-sourced generality. Therefore, a proof and a source are needed for the guesswork in VI. The production of all non-primitives from all the even series, without multiplication, is an important theorem. Dickson (and the other academics) must also have had a reason for refraining from putting multiplication within their equations to inform readers. This can only be because it would appear to them to be not generally possible “by definition” or guesswork, they would also need a proof. Subject to any further, early (hopefully mathematical) comments, I will arrange a new entry in the near future. Hoarwithy (talk) 14:32, 3 September 2011 (UTC)


 * Euclid and Dickson have proved that their equations produce ALL Pythagorean triples. The proofs are easy enough to follow and you can find them with little effort if you really want to understand them.  However, it appears you have some other agenda.  — Preceding unsigned comment added by 184.153.109.223 (talk) 06:09, 5 September 2011 (UTC)

If you are referring to Euclid’s equation in the write-up in Entries I and II, “Pythagorean Triple”, this merely suggests non-primitives can be obtained by separately multiplying a b c, in primitives $$a^2 + b^2 = c^2$$, by a multiplier $$k$$. This was referred to in my original XIII entry as the only way non-primitives could be produced from the standard equation results, is also not appropriate to this equation because m and n produce some non-primitives, is not mentioned by Dickson or the other Professors, and is not referenced to Euclid.

I have not found any proof that Dickson’s equations produce non-primitive triples. You say Dickson and Euclid proved that their equations produce ALL triples, and these are easily understood and found with little effort. I would be obliged if you could quote references to where this is claimed, and show me the values of m and n that produce the triples 9. 12. 15, and 15. 20. 25. This information would settle this dispute. If you can’t then I would ask you to reinstate that section of my work extending Dickson’s work, which you deleted. Shall we say two weeks from now (September 22) would be an adequate time scale for this? Hoarwithy (talk) 13:35, 8 September 2011 (UTC)


 * [9, 12, 15] and [15, 20, 25] can be easily obtained from Euclid using m = 2, n = 1, k = 3 for the first one, and m = 2, n = 1, k = 5 for the second. This is clearly explained and sourced in the Wikipedia article on Pythagorean triples, along with an explanation for the the need to introduce parameter k when generating ALL triples as opposed to just the primitives. But as Euclid well knew, it is enough to consider only the set of primitive triples (all of which are generated by the equation you cite), since ALL non-primitive solutions can be generated trivially from the primitive ones. In addition to all of the primitives, the version of the equation using only parameters m and n produces an infinite number of non-primitive triples of the form [ak,bk,ck] where k > 1 is a square or twice a square. To get only the primitives, m and n must be coprime, with m > n, and one of m,n must be even.


 * [9, 12, 15] is just as easily produced using Dickson's equations, as the example in VI clearly shows ( r = 6, s = 3, t = 6). You keep saying that you can find no proof that "Dickson’s equations produce non-primitive triples". This single example should be proof enough! Your other example, triple [15, 20, 25] is also easily produced using (r = 10, s = 5, t = 10). Nowhere in the source you cite does Dickson limit himself to the primitives (or non-primitives) as you have repeatedly and inaccurately claimed.  This is because (as he well knew)  his equations apply to both cases! To get as many non-primitives using Dickson's equations as we want, we need only require that factor pairs ( s and t ) share a common factor, and begin with r = 2. A source for Dickson's proof? For starters you can look at his own footnote (34) on page 169 of the book you have cited.  See also P.G. Egidi, D. Gambioli, A. Bottari, and H. Shotten [Footnotes 35, 36, 39, and 36a on the same page]. 


 * I assume you agree that Dickson's equations produce all the primitives. This is true only when s and t are coprime as in Line 1 below.  (If they were not coprime, then we could remove common factor k as shown on Line 2. (Clearly, if k is a factor of s and t, it is also a factor of r). But if instead we apply common factor k to our primitive triple(s) [ x, y, z], we get the non-primitive triple(s) [ x', y', z] as shown on Line 3 where r', s', t'  share common factor k > 1. '
 * $$\begin{array}{*{35}{l}}

1) & x=\text{  }(r+s),\text{          }y=(r+t),\text{        }z=(r+s+t)  \\   2) & {x}'=(r+s)k,\text{       }{y}'=(r+t)k,\text{        }{z}'=(r+s+t)k  \\ 3) & {x}'=({r}'+{s}'),\text{      }{y}'=({r}'+{t}'),\text{         }{z}'=({r}'+{s}'+{t}')  \\ \end{array}$$


 * Thus, given an arbitrary primitive [ x, y, z] with s, t coprime, we get all of its non-primitive multiples too. The latter have the form [ x', y', z'  ] where r', s', t share common factor k > 1.'

I will clarify the position - In July you arbitrarily deleted my entry XIII, which showed both source and proof that all Pythagorean triples, including non-primitives, can be calculated uniquely from the factors of each and every even number squared. This included a new and necessary algebraic extension to an equation first published by Professor Dickson. You substituted as a joint site your own anonymous, un-proved entry, with its single example, now VI.

You claimed there is an earlier proof to mine, which “can be found with little effort” saying, above, that Dickson’s own footnote (34) refers to his proof that his equation produces all non-primitives. This is incorrect. In his Amer. Math. Monthly, 1, 1894, 8, article he specifically confines himself to primitives. For the other references you quote, when providing them Dickson says these give the same rule as his.

Dickson says his equation is equivalent to Euclid’s. You had earlier claimed Euclid’s equation produces all the non-primitives, but this was also incorrect because, in your first paragraph above, you now agree a separate multiplier (k) needs to be applied to the x y z (varied to a b c) of his produced primitives. (In your opening sentence you appear to be suggesting k can be applied to m and n, but this would also be incorrect.) I pointed out that Euclid’s equation does produce a tiny minority of non-primitives, being a classic case that, in mathematics, examples are never taken as proofs. Yet, in your second paragraph above you repeat your assertion that a “single example should be proof enough”, which is mathematically incorrect.

You explain and give equations 1 – 3 above to show Dickson’s xyz and str can be multiplied by k to give non-primitives. Your earlier footnote to the above, removed recently, appeared to claim first publication for yourself. Yet Section F, point 7 of my website (http://www.calculatingpythagoreantriples.org.uk) referred to in my entry XIII, which you deleted has as its final conclusion “X Y Z A B G are equally affected by multiplication and division”. The A B G in my equations being the s t r in Dickson’s equations.

You have had two months to demonstrate in discussion an earlier proof to mine that all triples can be calculated from Dickson’s equations, and that they can be calculated from the factors of the even numbers $$r^2/2$$. You have failed to do so, just as I, and others, had failed to find one. I propose shortly replacing the combined entry VI with another. May I suggest that deleting or changing another mathematicians website without discussion is, at best, vandalism. It would have at least been courteous to raise your comments in discussion before arbitrary, anonymous censorship. Hoarwithy (talk) 21:27, 1 October 2011 (UTC)


 * I will not discuss this further. However, be advised that your webpage does not meet Wikipedia's standards as a source or reference.  Adding it to these pages will be considered vandalism.  — Preceding unsigned comment added by 184.153.109.223 (talk) 17:02, 3 October 2011 (UTC)

How strange you no longer wish to discuss this. In much the same way you took it upon yourself to delete a section without discussion, and when asked for justification have been unable to come up with any despite claiming it is readily available. This surely suggests it is you who are guilty of vandalism and are hardly in a position to accuse others of it. Hoarwithy (talk) 22:52, 3 October 2011 (UTC)

Combine IV and V?
Suggestion: Sections IV and V might be combined into a section called "Variations on Euclid"? — Preceding unsigned comment added by 184.153.109.223 (talk) 06:06, 1 October 2011 (UTC)

weird TeX
Someone who worked here extensively wrote a lot of stuff like this:
 * {{a}^{2}} + {{b}^{2}} = {{c}^{2}}

within math tags, instead of
 * a^2 + b^2 = c^2

and did a bunch of other things that look as if they were done by a lunatic or at least someone not particularly familiar with TeX. Sometimes curly braces are necessary (e.g. if you have more than one character in a subscript or a superscript, such as a_{2n}. But sometimes they're at best clutter.  Is there some reason why it was done that way? Michael Hardy (talk) 23:11, 11 October 2011 (UTC)


 * Heh heh, I can't claim credit for most of them, but in one edit in May 2010 I did put in a couple of instances of (1^{2}+2^{2}) etc., which were subsequently augmented by someone else to ({{1}^{2}}+{{2}^{2}}) etc. When you simplified the others you missed mine, so I've done them now. Don't worry about me -- I've recently upped my dosage of lunatic meds! Duoduoduo (talk) 19:43, 19 October 2011 (UTC)

Deleting my contribution
I made a contribution to this article little more than a week ago on "the universal set of Pythagorean triples." Someone removed it. Why? I made reference to an article that contains rigorous proof of a matrix of Pythagorean triples that form the universal set of triples. This is the ultimate in finding the triples. Why was it removed, and what must I do right this time to have it maintained? Should I rather write my own article, independent on this one, for finding any Pythagorean triple? — Preceding unsigned comment added by 119.46.251.60 (talk) 07:59, 6 December 2013 (UTC)
 * Please see the page WP:OR. The short summary is that Wikipedia does not publish original research, and that's what you were adding to the page.  (Also, for what it's worth, your method is a relatively minor variation of some of the formulas on this page; for example, if we set x = m/n and rescale then your triple is the same as the "usual" triple from Euclid's formula.) --JBL (talk) 19:35, 6 December 2013 (UTC)

Generating triples when one side is known
This section is wrong so someone should fix it:) Euclid's formula is designed to generate only primitive triples, but the text claims that it will generate all of them. Omitting the restriction that m and n should be coprime allows some non-primitive triples to be generated, but not all of them. E.g. in the example given with b = 24, (18, 24, 30) and various others are missed. In order to generate all possible triples, the factor k as in http://en.wikipedia.org/wiki/Pythagorean_triple#Generating_a_triple needs to be included. — Preceding unsigned comment added by 97.77.50.20 (talk) 04:30, 7 February 2014 (UTC)

Article impossible to understand if you don't know the topic already
The properties of this progression are: (a) the whole numbers are those of the common series and have unity as their common difference; (b) the numerators of the fractions, annexed to the whole numbers, are also the natural numbers; (c) the denominators of the fractions are the odd numbers, $$3,\text{ }5,\text{ }7,\text{ }9,$$ etc.


 * What is a common series?
 * What is unity?
 * What is a common difference?
 * What does annexing mean?

91.83.139.187 (talk) 07:40, 22 January 2019 (UTC)


 * — UnladenSwallow (talk) 19:30, 6 November 2021 (UTC)