## S000774

Primes p for which the polynomial (x^59 - 1)/(x - 1) mod p is irreducible.

2, 11, 13, 23, 31, 37, 43, 47, 59, 61, 67, 73, 83, 89, 97, 101, 103, 109, 113, 131, 149, 151, 157, 173, 179, 191, 211, 227, 229, 233, 269, 283, 313, 337, 347, 349, 367, 397, 401, 409, 419, 421, 431, 443, 457, 463, 467, 503, 509, 541, 563, 569, 571, 587, 601

1

This is the 58th-degree cyclotomic polynomial. These primes p satisfy the congruence p mod 59 == {0, 2, 6, 8, 10, 11, 13, 14, 18, 23, 24, 30, 31, 32, 33, 34, 37, 38, 39, 40, 42, 43, 44, 47, 50, 52, 54, 55, 56}.  The fraction of all primes in this sequence is 14/29. If we plotted n versus primepi(s(n)), then the plotted points would be very close to the line having slope 29/14.

T. D. Noe, Plot of 1000 terms

T. D. Noe, Table of 1000 terms

Wikipedia, Cyclotomic polynomial

(Mma) t = {}; n = 59; p = 1; While[Length[t] < 100, p = NextPrime[p]; If[Length[FactorList[(x^n - 1)/(x - 1), Modulus -> p]] == 2, AppendTo[t, p]]]; t

nonn

T. D. Noe, Dec 01 2015