A307590 a(n) is the smallest base b such that q = b^n - b^m + 1 is prime, where m = A276976(n).

2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 4, 2, 2, 2, 8, 2, 2, 3, 2, 14, 11, 2, 11, 29, 11, 5, 19, 14, 6, 27, 2, 3, 21, 8, 7, 10, 3, 4, 2, 14, 3, 5, 106, 3, 2, 44, 4, 3, 43, 4, 4, 21, 6, 16, 25, 41, 3, 12, 14, 10, 2, 3, 81, 28, 27, 66, 37, 17, 61, 5, 22, 12, 179, 197, 49, 2, 132, 178, 11

Offset: 1

Author

Thomas Ordowski, Apr 19 2019

Keywords

• nonn

Comments

If p is a prime, then a(p) is the smallest base b such that q = b^p - b + 1 is prime. These primes q == 1 (mod p) by Fermat's Little Theorem. Note that if p is a prime, then a(p) = 2 if and only if 2^p - 1 is prime, so p is a Mersenne exponent in A000043. Composite numbers n such that a(n) = 2 are 4, 6, 8, 10, 12, 14, 16, 22, 39, 45, 76, ... Cf. composite terms in A307625. Except 8, are these the same numbers?

a(80) does not exist because A276976(80) = 4 and b^8-b^4+1 is a factor of b^80-b^4+1. Similarly, a(n) also does not exist for n = 84, 160, 312, 320, 400, 588, 640, 684, 800, ... - Giovanni Resta, Apr 24 2019

Examples

			a(9) = 5 so the number 5^9 - 5^3 + 1 is a prime q == 1 (mod 9).
		

Crossrefs

Cf. A000043, A276976, A307625.

Programs

Mathematica

fQ[n_, m_] := AllTrue[Range[2, n - 1], PowerMod[#, m, n] == PowerMod[#, n, n] &]; f[1] = 0; f[2] = 1; f[n_] := Module[{m = 0}, While[!fQ[n, m], m++]; m]; a[n_] := Module[{b = 2, m = f[n]}, While[!PrimeQ[b^n - b^m + 1], b++]; b]; Array[a, 79] (* Amiram Eldar, Apr 19 2019 *)

PARI

a276976(n)=if(n<3, return(n-1)); forstep(m=1, n, n%2+1, for(b=0, n-1, if(Mod(b, n)^m-Mod(b, n)^n, next(2))); return(m)); \\ A276976
a(n) = my(b=2); while (!isprime(b^n - b^a276976(n) + 1), b++); b; \\ Michel Marcus, Apr 21 2019

Formula

q == 1 (mod n).

Extensions

More terms from Amiram Eldar, Apr 19 2019