## S000780

Primes p for which the polynomial (x^83 - 1)/(x - 1) mod p is irreducible.

2, 5, 13, 19, 43, 47, 53, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 137, 139, 149, 157, 163, 179, 181, 211, 223, 233, 239, 251, 257, 263, 269, 271, 281, 283, 307, 311, 337, 347, 367, 379, 389, 421, 433, 439, 449, 457, 461, 467, 487, 491, 503, 541, 569, 571

1

This is the 82nd-degree cyclotomic polynomial. These primes p satisfy the congruence p mod 83 == {0, 2, 5, 6, 8, 13, 14, 15, 18, 19, 20, 22, 24, 32, 34, 35, 39, 42, 43, 45, 46, 47, 50, 52, 53, 54, 55, 56, 57, 58, 60, 62, 66, 67, 71, 72, 73, 74, 76, 79, 80}. The fraction of all primes in this sequence is 20/41. If we plotted n versus primepi(s(n)), then the plotted points would be very close to the line having slope 41/20.

T. D. Noe, Plot of 1000 terms

T. D. Noe, Table of 1000 terms

Wikipedia, Cyclotomic polynomial

(Mma) t = {}; n = 83; 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

© Tony D Noe 2014-2015