A113517 Least k such that k^n-k+1 is prime, or 0 if there is no such k.

2, 2, 3, 2, 3, 2, 0, 3, 4, 4, 4, 2, 0, 5, 18, 2, 12, 2, 0, 7, 3, 11, 13, 7, 0, 167, 15, 6, 63, 2, 0, 7, 6, 21, 49, 3, 0, 27, 30, 3, 22, 106, 0, 10, 30, 4, 294, 7, 0, 32, 19, 6, 7, 41, 0, 21, 4, 14, 34, 2, 0, 12, 13, 6, 147, 37, 0, 14, 139, 22, 46, 179, 0, 4, 75, 69, 15, 11, 0, 5, 211, 130

Offset: 2

Author

T. D. Noe, Jan 12 2006

Keywords

• nonn

Comments

a(n) is 0 for n=8,14,20,... (n=2 mod 6) because, for those n, the polynomial x^n-x+1 has the factor x^2-x+1. Using a result of Selmer, it can be shown that x^n-x+1 is irreducible for all other n. Does a(n) exist for all n>1?

Crossrefs

Cf. A113516 (smallest k such that n^k-n+1 is prime).

Programs

Mathematica

Table[f=FactorList[x^n-x+1]; If[Length[f]>2, k=0, k=1; While[ !PrimeQ[k^n-k+1], k++ ]]; k, {n, 2, 100}]