A113516 Least k such that n^k-n+1 is prime.

2, 2, 2, 5, 2, 2, 13, 2, 3, 3, 5, 2, 3, 2, 2, 11, 2, 3, 17, 2, 2, 17, 4, 2, 3, 9, 2, 33, 7, 3, 7, 4, 2, 3, 5, 67, 5, 2, 9, 3, 2, 4, 25, 3, 4, 5, 5, 24, 3, 2, 3, 21, 3, 2, 9, 3, 2, 11, 2, 5, 3, 2, 4, 19, 31, 2, 29, 4, 2, 3019, 2, 21, 51, 3, 2, 3, 2, 2, 9, 2, 169, 965, 3, 3, 29, 3, 2848, 9, 2, 2, 3

Offset: 2

Author

T. D. Noe, Jan 12 2006

Keywords

• nonn

Comments

k can never be 8,14,20,... (k=2 mod 6) because, for those k, n^k-n+1 has the factor n^2-n+1, which is >1 for n>1. Using a result of Selmer, it can be shown that the polynomial x^k-x+1 is irreducible for all other k. The term a(93) is greater than 60000. Does a(n) exist for all n>1?

Links

• Ernst S. Selmer, On the irreducibility of certain trinomials, Mathematica Scandinavica (1956), Volume: 4, page 287-302.

Crossrefs

A343589 gives the primes.

Cf. A113517 (smallest k such that k^n-k+1 is prime).

Programs

Mathematica

Table[k=1; While[ !PrimeQ[n^k-n+1], k++ ]; k, {n, 2, 92}]

PARI

a(n) = my(k=1); while (!isprime(n^k-n+1), k++); k; \\ Michel Marcus, Nov 20 2021

Python

from sympy import isprime
def a(n):
    k = 2
    while not isprime(n**k - n + 1): k += 1
    return k
print([a(n) for n in range(2, 71)]) # Michael S. Branicky, Nov 20 2021