S000771


Primes p for which the polynomial (x^43 - 1)/(x - 1) mod p is irreducible.

3, 5, 19, 29, 43, 61, 71, 73, 89, 149, 157, 163, 191, 227, 233, 241, 263, 277, 313, 331, 347, 349, 373, 421, 433, 449, 463, 491, 499, 503, 521, 571, 577, 587, 593, 607, 631, 673, 691, 743, 751, 757, 761, 829, 863, 929, 937, 1009, 1019, 1051, 1061, 1087, 1093

1

S000771

This is the 42nd-degree cyclotomic polynomial. These primes p satisfy the congruence p mod 43 == {0, 3, 5, 12, 18, 19, 20, 26, 28, 29, 30, 33, 34}.  The fraction of all primes in this sequence is 2/7. If we plotted n versus primepi(s(n)), then the plotted points would be very close to the line having slope 7/2.

T. D. Noe, Plot of 1000 terms

T. D. Noe, Table of 1000 terms

Wikipedia, Cyclotomic polynomial

(Mma) t = {}; n = 43; p = 1; While[Length[t] < 100, p = NextPrime[p]; If[Length[FactorList[(x^n - 1)/(x - 1), Modulus -> p]] == 2, AppendTo[t, p]]]; t

Cf. A045309A042993A045401S000762-S000782.

nonn

T. D. Noe, Dec 01 2015

© Tony D Noe 2014-2015