Talk:Parasitic number

Currently, this article only really talks about the smallest n-parasitic numbers, and doesn't talk about parasitic numbers in a base other than base 10. Anyone who could expand in these directions, please do so. Mango juice talk 14:25, 17 May 2006 (UTC)

Merged content per AfD
I've merged content from 105263157894736842 (number) per AfD, but since I don't usually edit mathematics-related articles on a regular basis, I hope someone who does can edit my additions appropriately. --Deathphoenix ʕ 14:51, 23 May 2006 (UTC)


 * When I created this article, I merged everything I could, so I undid what you added (but it's in the history if anyone wants to see). The fact that (that number) is the decimal expansion of 1/19, and what parasitic numbers are, is covered already, and there wasn't anything else to merge.  Mango juice talk 15:09, 23 May 2006 (UTC)
 * Thanks. Hmmm... I could be mistaken, but one of the actions we strongly suggest against taking on an article during an AfD (in the AfD guide) is to merge content because of potential problems with GFDL if the consensus is to delete the article in question. Still, the article was kept as a merge & redirect, so no harm done. :-) --Deathphoenix ʕ 14:02, 24 May 2006 (UTC)
 * Perhaps it would have been better if the article had been moved, and then edited to the first version I made? Mango juice talk 15:00, 25 May 2006 (UTC)
 * Perhaps. It doesn't matter now because the redirect is in place. You can make it a cut and paste of whatever version is in the original article's history now. --Deathphoenix ʕ 15:13, 25 May 2006 (UTC)

n = 5
I'm afraid the smallest 5-parasitic number really is 142857, rather than the 1020408163265030612244897959183673469387755 originally calculated. I need to work on the formula to absorb that information. &mdash; Arthur Rubin | (talk) 13:06, 9 June 2006 (UTC)

This is fascinating, actually. 5 is the only exception, I'm pretty sure. It seems to me the interesting thing about the case of n=5 is that 49 is square, unlike several of the other values (such as 19 and 29) which are prime. More importantly, the repeating decimal part of 1/7 is only 6 digits long, whereas the repeating decimal part of 1/49 is much longer. I don't think there's an exception for n=4; 39 isn't prime, but 1/39 has just as long a pattern as 1/13, and 1/3 is obviously not so useful here. 59 is also prime. 69 = 3*23, but again 1/23 has just as long a pattern as 1/69, and 1/3 won't work because we'll always get extra digits, not a rotation. 79 and 89 are also prime, so no issue there. In order to have a lower period, the parasitic number would need to be based on a fraction with a shorter period, and only a divisor of (10n-1) would make that happen. Mango juice talk 20:22, 9 June 2006 (UTC)

My previous edit
Arthur_Rubin undid all my work accusing that it is almost entirely incorrect without any proof while my work is an evidence that it is correct. I request for a third party / common opinion from others / arbitration on this matter, e.g. Gandalf61 --Ling Kah Jai (talk) 15:54, 10 September 2009 (UTC)
 * I didn't say it was entirely incorrect. It's just that the parts which are (1) correct and (2) appropriate for Wikipedia (as opposed to Wikibooks) are appropriate only elsewhere.  Still, I welcome review.  — Arthur Rubin  (talk) 16:25, 10 September 2009 (UTC)
 * On these three articles Repeating decimal, Cyclic number, parasitic number, wherever I go, you undid my work, do you think you are taking thing personal? --Ling Kah Jai (talk) 16:45, 10 September 2009 (UTC)
 * If you propose your restatements on the talk page before making them, perhaps you can get consensus. But I only see the correct portions as duplication.  I left some of the #See also, and linked some of the terms in the body; everything else seems inappropriate.  — Arthur Rubin  (talk) 17:06, 10 September 2009 (UTC)
 * If you point out where I am wrong, then I will agree to your deletion. But can you? If you wish to change the sentence, just go ahead but please don't behave destructively. --Ling Kah Jai (talk) 17:18, 10 September 2009 (UTC)
 * I organized the article into proper sections. You just undid it. Nothing but vandalism. I changed some sentence structure you did not even read them!--Ling Kah Jai (talk) 17:20, 10 September 2009 (UTC)
 * If there is a committee that I can report, I want to report your behavior.--Ling Kah Jai (talk) 17:21, 10 September 2009 (UTC)
 * As I said elsewhere, your approach is repetitive of the section above and not simpler. — Arthur Rubin  (talk) 17:46, 10 September 2009 (UTC)
 * Asked somebody else it is repeatetive. Neither did you ever mentioned that it is repetitive.--Ling Kah Jai (talk) 17:53, 10 September 2009 (UTC) You said they are wrong. Are they wrong? ask a third person's opinion! --Ling Kah Jai (talk) 17:54, 10 September 2009 (UTC)

Direct approach
Using $$\circ$$ for concatentation of digit strings, the definition of Parasitic number reads:
 * $$n \times (y \circ d) = (d \circ y)$$
 * $$x = (y \circ d)\,$$

Mathematically, that reads (no longer using " " as contatenation)
 * $$y = \frac {x-d} {10} = n x - d 10^{m-1}$$

Multiplying by 10, and separating:
 * $$\left(10^m-1 \right) d = \left(10 n - 1\right) x$$

Or:
 * $$x = d \frac {10^m-1}{10n-1}$$

Perhaps this should be in the main article, after fixing the notation to formally indicate concatentation. — Arthur Rubin (talk) 18:06, 10 September 2009 (UTC)


 * No objection to the inclusion. And the simplified approach shall be in as well. --Ling Kah Jai (talk) 05:20, 11 September 2009 (UTC)


 * Your "simplified" approach is much more complicated; and the part which is necessary is already included. — Arthur Rubin  (talk) 06:08, 11 September 2009 (UTC)

Direct approach generalized
For a k-digit right shift, the equation becomes:
 * $$n \times (y \circ d) = (d \circ y)$$
 * $$x = (y \circ d)\,$$

(where d is no longer a "digit", but a k-digit number)

Mathematically, that reads (no longer using " " as contatenation)
 * $$y = \frac {x-d} {10^k} = n x - d 10^{m-k}$$

Multiplying by 10k, and separating:
 * $$\left(10^m-1 \right) d = \left(10^k n - 1\right) x$$

Or:
 * $$x = d \frac {10^m-1}{10^kn-1}$$

To avoid leading 0's, $$d \ge 10^{k-1}n$$.

For a k-digit left shift:
 * $$n \times (d \circ y) = (y \circ d)$$
 * $$x = (d \circ y)\,$$

(where d is no longer a "digit", but a k-digit number)

Mathematically, that reads (no longer using " " as contatenation)
 * $$y = \frac {n x-d} {10^k} = x - d 10^{m-k}$$

Multiplying by 10k, and separating:
 * $$\left(10^m-1 \right) d = \left(10^k - n\right) x$$

Or:
 * $$x = d \frac {10^m-1}{10^k-n}$$

Note also that, regardless of the value of m, $$GCD\left(10^k,n)\right) | d.$$

Here, for the count to be correct, we have the additional condition $$y < 10^{m-k}$$, which corresponds to
 * $$d \frac {n 10^{m-k} - 1} {10^k-n} < 10^{m-k},$$ or (approximately)
 * $$ d n < 10^k-n.$$ or
 * $$ d < \frac {10^k}{n}-1.$$

To avoid leadling 0s, $$d \ge 10^{k-1}.$$

For k=1, this only has non-trivial solutions for:

n = 3, x = 142857 or 285714 (and repeats, of course)

— Arthur Rubin (talk) 08:20, 14 September 2009 (UTC)


 * Look at that maths! Look at my maths (there is an update to User:Ling_Kah_Jai/Cyclic_permutations. Please read it again)! Provided that:
 * my method is not wrong;
 * nor repetivie repetitive; and
 * it is verifiable;
 * then you shall not defy me to add in the method. You just wouldn't admit that deleting my edit was a mistake. sigh! And there are no other people posting their opinion! To satisfy your pride, I will just call it 'alternative approach'.--Ling Kah Jai (talk) 16:33, 14 September 2009 (UTC)Ling Kah Jai
 * Why replace perfectly sensible algebra (my approach) with "arithmetic with (implied) variables and examples"? Combined with your confusion of a digit sequence D with the rational number equal to $$0.\bar \mathrm D,$$ your new method, although possibly correct, is not usable.  — Arthur Rubin  (talk) 16:43, 14 September 2009 (UTC)
 * Furthermore, your division by 199, etc., doesn't explain why "199" should be used.  If you do that, then your section is not much more than an extensive instanciation of mine, with slightly different notation, and using the repeating decimal instead of the simple digit sequence that the section requires.  Perhaps some examples could be included, but your examples don't explain why they work. — Arthur Rubin  (talk) 17:04, 14 September 2009 (UTC)


 * Updated with general rules. Please check up with Talk:Repeating_decimal, there is a trap in your method that does not always produce solution. I do not intend to replace your method but to add a separate more exclusive approach. --Ling Kah Jai (talk) 18:03, 14 September 2009 (UTC)


 * I have derived so many illustrative solutions (numbers). How many have you? Can you work out the solution using the method above for 2-digit left shift with n = 2 (i.e. multiply by 2)--Ling Kah Jai (talk) 18:19, 14 September 2009 (UTC) --Ling Kah Jai (talk) 18:22, 14 September 2009 (UTC)


 * You're right; I made a mistake there. I've now added a correction.  However, you do not have a justification why your examples are correct and/or complete.  — Arthur Rubin  (talk) 18:37, 14 September 2009 (UTC)


 * Refer to my latest text and you shall have the key. Now you believe in me and need to rely on the theory from repeating decimal for the exceptional case?--Ling Kah Jai (talk) 18:39, 14 September 2009 (UTC)

Dyson number
Dyson number redirects to this article. Why? Warrickball (talk) 17:26, 13 October 2011 (UTC)
 * I added a couple of sources to the article explaining this name. —David Eppstein (talk) 18:36, 13 October 2011 (UTC)
 * This link: http://www.nytimes.com/2009/03/29/magazine/29Dyson-t.html?pagewanted=8&_r=2&sq=Freeman%20dyson%20table%20integer&st=cse&scp=1 explains how Dyson did not actually invent these numbers, but instead, when the problem was raised in his presence, he announced immediately that the 2-parasitic number had 18 digits, apparently without having heard the problem before. This was considered by some to be impressive enough to get his name attached. --Matt Westwood 11:45, 26 May 2012 (UTC)

Bernstein paper
It seems only appropriate that the article refer to the paper "Multiplicative twins and primitive roots" by Bernstein, which deals with the problem of finding all numbers such that moving the last digit to the first position is a multiple of the original number, in various bases. A k-parasitic number is there called a k-multiplicative twin; it says "It is the purpose of this paper to state explicitly all k-multiplicative twins. This will be achieved in the first two sections." I added a reference, but wasn't sure if anything should be added to the article's text. I do not have a copy of the paper. --kundor (talk) 21:10, 4 April 2014 (UTC)

Origin of the name
Why are they so named? CielProfond (talk) 20:15, 23 July 2022 (UTC)