Talk:Transposable integer

Calling for stopping the edit war
Arthur Rubin, please stop placing your formula which has not been proven to be exactly accurate. --Ling Kah Jai (talk) 07:29, 18 September 2009 (UTC)
 * Please remove your formula from the lead. Mine (after corrections) is precisely accurate, although it probably would be better reformulated in terms of repeating decimals.  Yours (in terms of fractions, not repeating decimals) still has some problems with edge effects; it needs some more work to verify that the number of digits isn't off by one.  — Arthur Rubin  (talk) 12:39, 18 September 2009 (UTC)


 * 1. I used repeating decimal theory from the start. Who mentioned that the repeating decimal theory was complicated and not entirely correct?
 * 2. As I have proven to you, you don't even have a formula to express the exceptional case when the the period of the repeating decimal is less than k + 1. Then how could you claim that your formula is entirely correct? If you wish:
 * place your formula in a different section and let me prove you wrong. Don't mingle with the lede section and make people very confusing with two sets of incompatible formula; or
 * create a completely new article with the proof on your formula plus check your formula against my examples that I have included in this article. Perhaps the sub-page under your user page is a good location for such article when it is not ready! If you can prove it, then I can consider merging them.
 * :3. On your query with respect to multiplication by $r/s$ component, my formula works alright and it can integrate or is compatible with other components. It is your formula which has problem dealing with it. --Ling Kah Jai (talk) 14:35, 18 September 2009 (UTC)


 * 1. You used fractional representations from the start.  Repeating decimal representation would start with the equation for (with spaces representing concatenation) n &times 0.dydydy… = 0.ydydyd….
 * 2. There isn't a problem with the period being less than k + 1; the denominator won't divide the numerator, or there's a carry.
 * And your representation should not be in the lead, unless the article is to be entitled Fractional representation of cyclic permutations of integers under multiplication. — Arthur Rubin  (talk) 16:03, 18 September 2009 (UTC)


 * You denied the repeating decimal theory in the first place. Now perhaps you have to admit it is indispensable! I have to thank you for your denial or otherwise I would not have written this article to prove a point to you! Now if you say you can dispense the condition of exceptional case when the the period of the repeating decimal is less than k + 1, show me a working example that layman or myself can understand for why there are no solutions for 2-digit shifting right left cyclically for n = 4, 12 and 34. --Ling Kah Jai (talk) 16:22, 18 September 2009 (UTC)--Ling Kah Jai (talk) 01:50, 19 September 2009 (UTC)
 * Please don't change the heading of my section if you cannot prove that you theory can do without it. There is still something lacking from what you call 'direct presentation' to make it perfect.--Ling Kah Jai (talk) 16:28, 18 September 2009 (UTC)
 * If you want to take out the following, please prove a point to me:
 * As mentioned above, there are no solutions when $1/F$ has a period less than (k + 1).
 * - --Ling Kah Jai (talk) 16:39, 18 September 2009 (UTC)
 * (ec) My theory, or the correct repeating decimal representation, can handle all the cases. Yours seems to be able to, but is clearly unnecessarily complicated with respect to the true repeating decimal notation (which I don't have time to properly complete at this time) and arguably more complicated than mine.
 * I'd accept "prinicple" for the moment, to avoid edit wars, as long as it's properly tagged "disputed". However, tagging within section titles is problematic.
 * As for the period being less than k, that's actually false. 142857142857 has a left shift of 8 when multiplied by 2, and $1/7$ has a period of 6.  Do you want to try again?  — Arthur Rubin  (talk) 16:43, 18 September 2009 (UTC)
 * Your formulation is also right. I apologize for the false claim that it could not cater for the exception cases. Can you formulate the third case?--Ling Kah Jai (talk) 06:05, 19 September 2009 (UTC)
 * It's difficult. The digit sequence D seems not to be the correct multiplier.  I'll see what I can do.  — Arthur Rubin  (talk) 07:41, 19 September 2009 (UTC)

Fractional multiples
I also suggest that the multiplication by $r/s$ component be removed from the lede, as the subsidiary conditions differ, depending on which of r or s is larger. — Arthur Rubin (talk) 13:00, 18 September 2009 (UTC)
 * OK, This formula for this component is slightly different. Updated.--Ling Kah Jai (talk) 15:24, 18 September 2009 (UTC)

Further alternative formulation
Writing the equations in terms of x = 0.XXX…, but not directly as fractions, may lead to a better formulation which we could agree on in the lede. It's certainly cleaner than your F. — Arthur Rubin (talk) 13:00, 18 September 2009 (UTC)
 * That would be a repeating decimal representation. — Arthur Rubin  (talk) 16:03, 18 September 2009 (UTC)

Concept And principle
Arthur Rubin, in my opinion, though direct algebra approach provides the solutions, it does not explain the occurrence of the cyclic permuted integers nor does it provide an an insight to in-depth understanding of this subject. The repeating decimal approach does all. Repeating decimal and fraction are together a single approach. There is no division line between repeating decimal and fraction approaches again. For this reason, I will change the first section heading again. --Ling Kah Jai (talk) 06:02, 26 September 2009 (UTC)
 * You are not using a repeating decimal representation. Direct manipulation of the actual repeating decimal, .XXX…, would be a repeating decimal representation.  You're starting with the fraction, instead.  — Arthur Rubin  (talk) 06:42, 26 September 2009 (UTC)

New condition added by Arthur Rubin
Arthur Rubin, can you test the condition added by you is valid against the following:
 * Cyclic left shift (or right shift) of 2 places when multiplied by 1. I would think 0101, 0202, ..., 9999 are all valid solutions (converted from 1/99 and thus, though they can be considered to be trivial).

It appears to me that your new condition have ruled out these valid solutions.--Ling Kah Jai (talk) 09:29, 29 September 2009 (UTC)


 * Multiplication by 1 is a special case, generally considered unacceptable in the traditional formulations, such as parasitic number. I think it should probably be considered unacceptable here, as well.  However, the article makes sense either way.  — Arthur Rubin  (talk) 09:33, 29 September 2009 (UTC)


 * I would think otherwise: the condition that you added may be redundant and unnecessary. Can you list arithmetic examples that require valid use of the new condition in addition to what we already have?--Ling Kah Jai (talk)


 * I believe you orginally added it (> k, rather than not dividing k) as a cross-check so that it could be quickly verified that certain combinations of k and n are impossible. I suppose it's not really necessary, if you don't mind checking all factors of F0.  — Arthur Rubin  (talk) 09:44, 29 September 2009 (UTC)
 * Yes I did that. But I found that your way works generally, i.e. to have a solution then n*j/F shall be less than 1. --Ling Kah Jai (talk) 09:51, 29 September 2009 (UTC)

Do not simply delete writing
Arthur Rubin, please do not simply delete writing as you write wish. If you think it is inaccurate, challenge it. There is no difficult mathematics involve, just elementary arithmetic. Common Come on, you are a PHD! If you think it does not belong here, move it! But I think it belongs here. Addition and subtraction of cyclic permutations belongs to repeating decimal? What a non-sense. Addition and subtraction of cyclic permutations is in accurate inaccurate? what a non-sense. The following is in accurate inaccurate? What a non-sense.


 * If j is a six-digit integer, its next cyclic permutation k can be obtained from:

k = 10j − 999999n Where n is the first digit of j


 * Example 157509 = 915750×10 - 9×999999.

--Ling Kah Jai (talk) 01:52, 9 October 2009 (UTC)


 * That's correct. It's not what you had said in the article, and it has nothing to do with repeating decimals or your fraction argument.  It's closer to supporting my argments.  — Arthur Rubin  (talk) 02:56, 9 October 2009 (UTC)


 * You really has forgotten the subject: cyclic permutation of integer! Not cyclic permutation of integer by multiplication! I proclaim your act as vandalism!. --Ling Kah Jai (talk) 03:07, 9 October 2009 (UTC)


 * You can proclaim whatever you want. — Arthur Rubin  (talk) 05:40, 9 October 2009 (UTC)

Today's nonsense additions
I'm going to give a paragraph-by-paragraph discusion of the massive edits which almost certainly do not belong in this article. Please wait to revert until I finish. — Arthur Rubin (talk) 02:59, 9 October 2009 (UTC)
 * You should wait to delete! Your argument first before any deletion!--Ling Kah Jai (talk) 03:08, 9 October 2009 (UTC)
 * Who said that users cannot have massive addition, so long as it is right?--Ling Kah Jai (talk) 03:10, 9 October 2009 (UTC)
 * (+ec) It's not right.
 * You blocked my line-by-line commentary with your inappropriate edits, so it will have to wait a day or two. I'm not up to editing that much inappopriate formulas.  I'm going to propose the article for deletion, instead.  Perhaps something can be salvaged that way.  — Arthur Rubin  (talk) 03:42, 9 October 2009 (UTC)
 * I don't see that there are many formulas. There is only one formula only. The rest is arithmetic. If there is any error in the arithmetic, is it so difficult to say so? Read before deletion. Your have just break the three revert rules and even deleted my discussion here.--Ling Kah Jai (talk)
 * I deleted the discussion in the hope that I could get my commentary in, but it still failed. I had about the same amount of text as you had in the misplaced lede.  I guess I'll just have to propose the article for deletion.  The name is still hopeless.  The "addition" section you added is just arbitrary cyclic permutations, as you don't explain why they're of any interest, and should be summarily deleted as trivia.  The first part has some credibilty (except for the 2/5 text, which should always have been in repeating decimal), but should be below the lede, possibly as a third alternative method.  The GCD is somewhat interesting, but it's more related to the direct method than the repeating decimal or fraction method, and possibly should be added to my section.  — Arthur Rubin  (talk) 04:05, 9 October 2009 (UTC)


 * My original addendum to the lede is not supposed to be attached to any method but is meant to be general for cyclic permutations.
 * Who decides that a section is trivia or otherwise? Provided it is something different, there is a right for the writing to be in. I have seen trivia writing and I do not think the addition and subtraction of cyclic permutations is trivia as there have not been any writing for it. --Ling Kah Jai (talk) 04:30, 9 October 2009 (UTC)
 * It's trivia and unsourced in that no reliable source mentions it. At least, reliable sources mention the cyclic-shift-by-multiplication problem.  We sometimes, in mathematics articles, have proofs which we cannot allocate to a source, but the theorem must have been mentioned somewhere.  — Arthur Rubin  (talk) 15:35, 9 October 2009 (UTC)
 * And the inappropriate lede addition consist of some general comments on cyclic permutations, which are mostly true (even though 9-repdigit is a WP:neolgism), but probably should be moved to Wikibooks, as there is no evidence that it isn't your sole work, and the 2/5 material which probably should be moved to repeating decimal. I would have used the term pure repeating decimal for one which has no cyclic term, but I can't find a reference, so I couldn't use it here in Wikipedia.  — Arthur Rubin  (talk) 15:38, 9 October 2009 (UTC)
 * If you need a source and a theorem, then cyclic group (of multiplication modulos n) shall be the source and theorem. It is in fact very simple if we link everything to groups and cyclic groups. --Ling Kah Jai (talk) 03:58, 12 October 2009 (UTC)
 * I also have to repeatedly say it aloud: Do you need to quote the source again to verify arithmetic works which is easily verifiable? --Ling Kah Jai (talk) 01:47, 14 October 2009 (UTC)

Group theory
Arthur Rubin, from your edit, I doubt you know anything about group theory. --Ling Kah Jai (talk) 01:38, 14 October 2009 (UTC)
 * The relevant parts of group theory are actually from monoid theory (the abstract theory of functions of one variable, which apparently is not called "monoid" here). — Arthur Rubin  (talk) 16:08, 14 October 2009 (UTC)

Merger proposal
Proposal to merge: Cyclic number into transposable integer; dated: November 202. Proposer's Rationale: ''To a layperson, transposable integers and cyclic numbers look very similar. The article states, "These specific integers, known as transposable integers, can be but are not always cyclic numbers. The characterization of such numbers can be done using repeating decimals (and thus the related fractions), or directly." without describing the difference.'' Discuss here. — Preceding unsigned comment added by GenQuest (talk • contribs) 21:12, 13 November 2023 (UTC)
 * Oppose; similar, but different. Readers are best served by keeping these well-developed articles distinct. They are already appropriately linked. Klbrain (talk) 20:12, 2 December 2023 (UTC)
 * Oppose: According to the definitions given in these articles these are different concepts. A "cyclic number" is a particular type of "transposable integer". Both articles could use some cleanup and maybe better sourcing, but I don't see the benefit of merging them. –jacobolus (t) 15:40, 29 May 2024 (UTC)