Talk:Cyclic number

Suggest to merge this article with repeating decimal
Ling Kah Jai (talk) 12:47, 30 September 2008 (UTC)
 * Main article and sub-article relationship has been established.--Ling Kah Jai (talk) 13:46, 10 September 2009 (UTC)
 * No, they haven't, and there's no reason to merge. — Arthur Rubin  (talk) 15:30, 10 September 2009 (UTC)

Reaman Numerals?
In 1953, I (independently) discovered what I called a "Reaman Numeral" (after my father's first name). As you will see, it has 44 digits: 10,112,359,550,561,797,752,808,988,764,044,943,820,224,719. Its derivation is available at http://niquette.com/puzzles/reaman1p.htm, and other Reamans are derived at http://niquette.com/puzzles/reaman2p.htm.

Now, I am not a mathematician. What I want to know is surely obvious: Have numbers with this property been discovered by others? If so, what are they called? If not, I would be delighted if wikipedia were to give recognition to Reaman Numerals, including links as given above. Paul Niquette 23:59, 19 December 2006 (UTC)


 * Paul, you may now refer to Repeating decimal which gives you hint on the period of the repeating decimal once you know the denominator.--Ling Kah Jai (talk) 14:55, 15 May 2009 (UTC)


 * I think they might be called "parasitic numbers." Remove the commas and enter the numeral into the OEIS search box. Two results come up:


 * A081463 Numbers which when multiplied by their final digit have products with same digital sequence except that last is first. Numbers obtained by concatenating a term any number of times with itself also have the defining property and are omitted.


 * A092697 Least n-parasitic number.


 * I would prefer "Reaman" to "parasitic," but it might be too late to do anything about it. Back in 1953 you might have been able to change the tide. PrimeFan 22:11, 20 December 2006 (UTC)
 * Thanks for the information. I will chase down the provenance for dates and add an epilog preceded by "also known as 'parasitic numbers'..." on the solution page.  Now, what are your thoughts about an External link to my site? Paul Niquette 04:04, 21 December 2006 (UTC)
 * It's good that you suggested to someone else to add it as opposed to adding it yourself. The most important thing to avoid here is the appearance of linkspam. I will carefully look over your site and I will also put the question to the WP:NUM project members. In the meantime, merry Christmas! PrimeFan 20:51, 21 December 2006 (UTC)
 * Please accept my gratitude for your diligence in addressing my suggestion. I shall be glad to receive any guidance you and others might care to make.  Best wishes for 2007 -- and beyond. Paul Niquette 13:53, 22 December 2006 (UTC)
 * Following up on your most recent entry here: I have studied parasite numbers as you suggested and, yes, "parasite numbers" do indeed map to "Reaman Numerals" but with an entirely different derivation -- a derivation that requires division by extraordinarily long decimal representation of a fraction and therefore arduous indeed before the advent of calculators. My method requires manual long division of an extendible dividend by a fixed integer.  All of the published dates that I have found associated with "parasite numbers" are in the late 20th and early 21st century; my original solution (the first Reaman Numeral with its "do-si-do digit" of 9) dates back to 1953, as described in the "Historical Note" at http://www.niquette.com/puzzles/reaman1s.htm. My great-grandchildren and I await your judgment on whether an external link would add value to the "parasite numbers" article (or some other) in Wikipedia. Paul Niquette 00:29, 8 January 2007 (UTC)

A year and a half later, my great-grandchildren and I are still waiting for a reply to my previous follow-up, hoping with ego at the ready for a favorable verdict and thus recognition of my little 1953 discovery. Paul Niquette (talk) 21:34, 9 July 2009 (UTC)

Number patterns
Taking the last digits of the multiplication table combining them with the digits that sum up to 10 give a direct and an opposite direction. For example given a number 10 and adding nine to it,it gives 19,28,37,46,55,64,73,82,91, and if it was multiples of nine the answer would be 18,27,36,45,54,63,72,81,90. On one case it adds up to ten and on the other it adds up to nine. To see such proof you must divide 1 by 81 and multiply the answer beginning with one to 9 and see the pattern that takes place. A sequence for example reveals such description by dividing 1 by 999991 to 1 divided by 999999. Cyclic or permutable numbers are another examples.

One divided by 81=0.012345679..... a cyclic number with 10,19,28,37,46,55,64,73,1 as remainders
 * 12345679*1= 1 2 3 4 5 6 7 9
 * 12345679*2= 2 4 6 9 1 3 5 8
 * 12345679*3= 3 7 0 3 7 0 3 7
 * 12345679*4= 4 9 3 8 2 7 1 6
 * 12345679*5= 6 1 7 2 8 3 9 5
 * 12345679*6= 7 4 0 7 4 0 7 4
 * 12345679*7= 8 6 4 1 9 7 5 3
 * 12345679*8= 9 8 7 6 5 4 3 2
 * 12345679*9= 1 1 1 1 1 1 1 1

example of a cyclic number.


 * 1/17=0.058823529411764705882352941176470....
 * 16 repeated remainders:10,15,14,4,6,9,5,16,7,2,3,13,11,8,12,1
 * Difference from 16 to 7 of the remainders always nine on a cyclic # except 1/7.
 * Adding the first eight remainders to the second half, always equals 17.
 * Repeated quotient:0588235294117647. If from 0 to 5 of the quotient is positive
 * then from 9 to 4 is negative 5.
 * Multiplying each remainder by the quotient gives a cyclic number.
 * The sum of the quotient:72
 * The sum of the remainders:136
 * 136/17=8-72/9=8
 * 0588235294117647*136=79999999999999992
 * 1/17*136=8

Only works when the quotient has a length of an even number.

Interesting question to know why numbers share between themselves and how they share. When they are given a space between themselves they don't share. example
 * 1/91=0.010989010989
 * 1/991=0.0010090817356...
 * 1/9991=0.000100090081072965669....

21:24, May 24, 2007 user twentythreethousand


 * Have you tried looking at this in bases other than base 10? PrimeFan 22:48, 24 May 2007 (UTC)


 * I have tried others base math, the only similarities I've found is with sexagesimal
 * With Babylonian mathematics you'll be dealing with 60 and 59 and with our base ten digits it would be ten and nine.
 * example:
 * 1/666=5, 24, 19, 27, 34, 3, 14, 35, 40, 32, 25, 56, 45
 * remainders=  270,216,306,378,36,162,396,450,360,288,630,504
 * 24+35=59,19+40=59,27+32=59,34+25=59,3+56=59,14+45=59,
 * sum of quotients 59*6=354
 * 270+396=666,216+450=666,306+360=666,378+288=666,36+630=666,162+504=666,
 * sum of remainders 666*6=3996

Twentythreethousand (talk) 22:23, 29 January 2008 (UTC)

Cyclic permutation of integer
It has been proposed that Cyclic permutation of integer be merged with Cyclic number. I don't recall who proposed it, but a discussion section wasn't set up, until now. — Arthur Rubin (talk) 16:28, 23 September 2009 (UTC)

Actually, I think Cyclic permutation of integer should be merged with Parasitic number, if an appropriate name can be determined. It seems more appropriate than Cyclic number as a target. — Arthur Rubin (talk) 16:28, 23 September 2009 (UTC)

Trivial cases? And more info requested
The article says that:


 * The following trivial cases are typically excluded:
 * single digits, e.g.: 5
 * repeated digits, e.g.: 555

But those aren't cyclic numbers anyway, are they?

Also, the article isn't clear on how long of a run it required to call it a cyclic number - does it have to generate all permutations in order to be considered cyclic? Or just some? 212.9.31.12 (talk) 08:34, 26 November 2013 (UTC)
 * Re 5 & repeated digits - this should be thrown out. Or do I miss something? --User:Haraldmmueller 14:41, 7 June 2018 (UTC)

Unclear point in the section "Properties of cyclic numbers"
The following sentence currently appears under Properties of cyclic numbers: "All cyclic numbers are divisible by 'base−1' (9 in decimal) and the sum of the remainder is the a multiple of the divisor. (This follows from the previous point.)" In addition to it containing an obvious grammatical error... what exactly does it mean? Specifically, the sum of what remainder is a multiple of what divisor? 2A02:8109:9340:136C:309A:2996:B95D:ACF7 (talk) 00:24, 7 June 2014 (UTC)

I traslated article in russian and I think also, that sentence is not clear. I have to remove it from translation to russian. Jumpow (talk) 15:11, 11 April 2017 (UTC)

Representation of digits in base 12 and other bases
I recently "fixed" line 70, where I replaced the 'Ɛ' in this number '076Ɛ45076Ɛ45076Ɛ45' by a 'B', because when using a base greater than 10, it is common practice to use letters A through Z... However, I am now just finding out that there is a notation specifically for duodecimal where ᘔ represents 10, and Ɛ represents 11. It's nice to know that specific symbols exists for base 12, but it seems to me that it might be confusing to use them in one case (base 12), and use letters for any other base greater than 10 in the same page. 'Ɛ' has previously been removed by anonymous with no explanation, probably because it seemed out of place, and then put back by Arthur Rubin without a real explanation. I don't want to revert back my change without first asking other editors what they think. Please comment. Dhrm77 (talk) 03:22, 22 November 2015 (UTC)

Where to find Special Cases section?
Read in article: (all successive multiples being cyclic permutations) listed in the Special Cases section above)

Where is that Special Cases section? Jumpow (talk) 17:49, 11 April 2017 (UTC)

and the sum of the remainder is the a multiple of the divisor.
I'm not sure I understand "and the sum of the remainder is the a multiple of the divisor" (my emphasis) I'd have thought that the remainder was always less than the divisor, but in any event it can't be both the and a, can it?  Ϣere Spiel  Chequers  14:53, 2 March 2018 (UTC)

Cyclic numbers and digital period of primes
It says that "if the digital period of 1/p (where p is prime) is p − 1, then the digits represent a cyclic number". This is incorrect. It is only if p is a maximal prime number, i.e. 7,17,19, 23, 29, 47... For all other prime numbers, the period is not cyclic as defined in the previous section 'Details'. In fact, this section explicitly mentions the number 076923, which is the period of 1/13, as not being cyclic. Proton148 (talk) 07:20, 17 October 2023 (UTC)


 * Yes and the period of 1/13 is 6, not 12 (which is 13 - 1). --JBL (talk) 19:57, 17 October 2023 (UTC)