Talk:Permutable prime

Note to PrimeFan: Do you dispute that primutation is a reasonable synonym for a Permutable prime? I added primutation as a synonym to the Permutable prime page in good faith, not as a joke or as defacement. I respect the work you've done sharing your extensive knowledge of the many varieties of prime numbers, and I will not waste your time with a edit/re-edit war. I do however ask that you reconsider the merits of primutation, and not just rely on Google-ability as a sole standard of legitimacy. Google does not contain all of human knowledge, and specifically cannot be trusted to know about all the forms of Technical terminology in use by various communities. Primutation is a real piece of technical terminology in use by certain communities of mathematical enthusiasts and deserves to be acknowledged as such, despite not having found its way into web publication thus far.

Thanks for your consideration (and keep up the good work on prime-related pages!)

--71.56.181.243 13:29, 20 January 2006 (UTC) Markov


 * You make some good points. You're right that "Google does not contain all of human knowledge." In this case, Google might not contain a link to a professional mathematician's paper which uses the term "primutation" to mean "permutable prime," because maybe that paper has appeared in a print journal but not online. And if there is such a paper by professional mathematician, please re-add "primutation" back to the article, along with a bibliographic note in either APA or MLA style. PrimeFan 00:42, 21 January 2006 (UTC)

Contradiction
The article currently contains the statement

"It is proved that no permutable prime exists which contains three different of the four digits 1, 3, 7, 9, as well as that there exists no permutable prime composed of two or more of each of two digits selected from 1, 3, 7, 9."

The second part of this statement is wrong, a counterexample is for example 337, which clearly consists of two digits selected from 1, 3, 7 and 9. Unfortunately I don't have access to the cited paper. I would appreciate if someone who has access could check if the person who added this to the article misread that source somehow or if there is a factual error in the cited paper. Toshio Yamaguchi (talk) 16:24, 23 January 2011 (UTC)


 * There is no contradiction. It correctly says "two or more of each of two digits". 337 only has one '7' so it is not a counterexample. The result is also mentioned in another form at the reference http://primes.utm.edu/glossary/page.php?sort=PermutablePrime. PrimeHunter (talk) 17:50, 23 January 2011 (UTC)


 * Agreed. That was a misunderstanding on my part. I missed the "of each" part when reading it. This sentence simply contains the word 'of' a bit too often (ironie intended) for my taste.Toshio Yamaguchi (talk) 19:31, 23 January 2011 (UTC)

near-repdigit
Begin OR:

If b is a primitive root modulo some prime p>2, then there are at most a finite number of permutable primes in base b which are not near-repdigits. (If Artin's conjecture on primitive roots holds, then this can be done for b not a square.)

Proof: If n is sufficiently large then there are at least 2(p&minus;1) of some digit x, and at least 2 other digits (either two ys, or a y and a z) in the number.

I claim that either $$\underbrace{xxx...x}_{p-1} yy$$ or $$\underbrace{xxx...x}_{2(p-1)} yz$$ can be permuted to take all values modulo p. In the first case, noting that bk takes all congruence classes other than 0 (mod p), for k in the range covered by the b&minus;1 "x"s, we can obtain differences of (u &minus; 1) (y &minus; x) and (u' &minus; b') (y &minus; x) for any nonzero u.

(u &minus; 1) omits only the congruence class &minus;1 (mod p), and (u' &minus; b') omits only the congruence class &minus;b' (mod p''), so no congruence class is omitted.

For the second case, we can obtain a difference of (u &minus; 1) (x &minus; z) + (v &minus; b) (x &minus; y) for any nonzero u and 'v''; The first term omits only (z &minus; x), and the second omits only b(y &minus; x); so no congruence class is omitted in the sum.

Emd OR

Although the above is original research on my part, I would be surprised if the inadequately indicated reference "(M. Fiorentini, 2015)" fails to make those statements. — Arthur Rubin (talk) 22:47, 22 December 2015 (UTC)