Talk:Unique prime number

Old post[edit]

This is not very understandable. How about giving some (1,2) examples with calculations? — Preceding unsigned comment added by 211.225.34.183 (talk • contribs) 12:40, 1 July 2005 (UTC)[reply]

Not explained[edit]

The original definition of the Unique prime depends massively on the base 10, apparently selected as being commonly used. Admittedly, the base 2 is also mentioned. — Preceding unsigned comment added by 204.68.119.201 (talk) 15:08, 24 April 2014 (UTC)[reply]

Because base 2 is the natural base, but base 10 is "artifical". — Preceding unsigned comment added by 180.204.20.183 (talk) 15:48, 8 October 2014 (UTC)[reply]

It will be interesting to see 180.204.20.183 prove that his words "natural" and "artificial" mean anything. — Preceding unsigned comment added by 92.26.7.194 (talk) 11:03, 14 October 2016 (UTC)[reply]

So is base 3 natural or artificial? How about the rest of the infinity of bases? 216.71.19.121 (talk) 20:02, 22 July 2017 (UTC)[reply]

Reference needed[edit]

Beside boring lists and tables, the main mathematical results of the article are

The characterization of prime periods, which is a corollary of Zsigmondy's theorem
The characterization of the primes of a given period as the prime divisors of $\Phi _{n}(b)/\gcd(\Phi _{n}(b),n)$

For the latter, no link nor reference are given. I am able to prove that the primes of period n are the prime divisors of $\Phi _{n}(b)$ that do not divide n (I have sketched the proof in a collapsed box), but I am unable to complete the proof. For this, one needs to bound the multiplicity of the prime factors of $\Phi _{n}(b)$ , and I do not know any standard proof method for that. In other words, I have proved that the primes of period n are the prime divisors of $\Phi _{n}(b)/\gcd(\Phi _{n}(b),n^{k})$ for k sufficiently large, but I am unable to prove that k = 1 always suffices.

I suspect that the claimed result has not really been proved, as, if it would be, this would probably provide a method for proving all the conjectures of the article that rely on multiplicities of factors.

So, please, provide a reference, and check it (or give me an access to it). D.Lazard (talk) 15:15, 16 October 2016 (UTC)[reply]

Still, I have not found a reference for the characterization of the primes of a given period as the prime divisors of

\Phi _{n}(b)/\gcd(\Phi _{n}(b),n),

but I am now convinced that it is true for every even base

b

(which includes the binary and the decimal cases). For an odd base, the period of 2 is 1, and 2 is a unique prime if and only if

b - 1

is a power of two. For an odd base, an odd prime

p

is a unique prime if and only if

\Phi _{n}(b)/\gcd(\Phi _{n}(b),n)=2^{k}p^{m},

for some

n, k, m

(over an odd base, 2 has no period length).

I'll try to implement this in the article. D.Lazard (talk) 13:17, 20 October 2016 (UTC)[reply]

There are datas in GitHub about generalized unique prime to various bases: sorted by period length and sorted by base ——2402:7500:916:2355:CDC1:1D1E:7F6D:4F46 (talk) 15:47, 17 September 2021 (UTC)[reply]

Is 360356 currently the longest known probable unique prime period[edit]

Although the repunit R₂₇₀₃₄₃ is listed as the largest known probable unique prime, the number 10¹⁸⁰¹⁷⁸+1/101 is a probable unique prime with reciprocal period 360356. Although it has fewer digits than R₂₇₀₃₄₃, it has a longer unique period.

Is the period 360356 of 10¹⁸⁰¹⁷⁸+1/101 the longest currently known unique prime period? luokehao, 15 January 2021, 12:59 (UTC)

No, it is 8177207, R_8177207 is the largest known probable prime, see PRP top. ——2402:7500:916:2355:CDC1:1D1E:7F6D:4F46 (talk) 15:44, 17 September 2021 (UTC)[reply]

Another reference about your question, including a list of known unique periods: [1] and [2] ——2402:7500:916:2355:CDC1:1D1E:7F6D:4F46 (talk) 15:51, 17 September 2021 (UTC)[reply]