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A306076 Bases in which 11 is a unique-period prime. 5

%I

%S 2,3,10,12,21,43,87,120,122,175,241,351,483,703,967,1330,1332,1407,

%T 1935,2661,2815,3871,5323,5631,7743,10647,11263,14640,14642,15487,

%U 21295,22527,29281,30975,42591,45055,58563,61951,85183,90111,117127,123903,161050,161052,170367,180223,234255,247807,322101,340735

%N Bases in which 11 is a unique-period prime.

%C A prime p is called a unique-period prime in base b if there is no other prime q such that the period length of the base-b expansion of its reciprocal, 1/p, is equal to the period length of the reciprocal of q, 1/q.

%C A prime p is a unique-period prime in base b if and only if Zs(b, 1, ord(b,p)) = p^k, k >= 1. Here Zs(b, 1, d) is the greatest divisor of b^d - 1 that is coprime to b^m - 1 for all positive integers m < d, and ord(b,p) is the multiplicative order of b modulo p.

%C b is a term if and only if: (a) b = 11^t + 1, t >= 1; (b) b = 2^s*11^t - 1, s >= 0, t >= 1; (c) b = 2, 3.

%C For every odd prime p, p is a unique-period prime in base b if b = p^t + 1, t >= 1 or b = 2^s*p^t - 1, s >= 0, t >= 1. These are trivial bases in which p is a unique-period prime, with ord(b,p) = 1 or 2. By Faltings's theorem, there are only finitely many nontrivial bases in which p is also a unique-period prime, with ord(b,p) >= 3. For p = 11, the nontrivial bases are 2, 3.

%H Jianing Song, <a href="/A306076/b306076.txt">Table of n, a(n) for n = 1..590</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Unique_prime">Unique prime</a>

%e 1/11 has period length 10 in base 2. Note that 3, 11, 31 are the only prime factors of 2^10 - 1 = 1023, but 1/3 has period length 2 and 1/31 has period length 5, so 11 is a unique-period prime in base 2.

%e 1/11 has period length 5 in base 3. Note that 2, 11 are the only prime factors of 3^5 - 1 = 242, but 1/2 has period length 1, so 11 is a unique-period prime in base 3.

%o (PARI)

%o p = 11;

%o gpf(n)=if(n>1, vecmax(factor(n)[, 1]), 1);

%o test(n, q)=while(n%p==0, n/=p); if(q>1, while(n%q==0, n/=q)); n==1;

%o for(n=2, 10^6, if(gcd(n, p)==1, if(test(polcyclo(znorder(Mod(n, p)), n), gpf(znorder(Mod(n, p)))), print1(n, ", "))));

%Y Cf. A040017 (unique-period primes in base 10), A144755 (unique-period primes in base 2).

%Y Bases in which p is a unique-period prime: A000051 (p=2), A306073 (p=3), A306074 (p=5), A306075 (p=7), this sequence (p=11), A306077 (p=13).

%K easy,nonn

%O 1,1

%A _Jianing Song_, Jun 19 2018

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Last modified March 5 11:57 EST 2021. Contains 341823 sequences. (Running on oeis4.)