A383237 - OEIS

%I #11 Apr 24 2025 13:22:49

%S 2,29,41,47,71,131,179,197,233,239,257,269,311,353,443,461,491,509,

%T 587,647,653,683,761,857,863,887,929,947,1013,1061,1223,1277,1283,

%U 1289,1301,1361,1373,1409,1427,1439,1499,1511,1559,1619,1637,1733,1823,1973,1979

%N Primes p such that x^5+x+1 has no roots modulo p.

%C Every term is congruent to 2 modulo 3, hence, except for a(1) = 2, to 5 modulo 6.

%H Jayde S. Massmann, <a href="/A383237/b383237.txt">Table of n, a(n) for n = 1..1000</a>

%e a(1) = 2 because 0^5+0+1 = 1 and 1^5+1+1 = 3; neither is 0 mod 2.

%e a(2) = 29, as for p = 3, 5, 7, 11, 13, 17, 19, 23, x^5+x+1 has a root modulo p, namely 1, 2, 2, 9, 3, 10, 3, 15, respectively.

%o (PARI) isok(p) = if (isprime(p), !#polrootsmod(x^5+x+1, p)); \\ _Michel Marcus_, Apr 20 2025

%Y Subsequence of A003627.

%K nonn

%O 1,1

%A _Jayde S. Massmann_, Apr 20 2025