A065902 Smallest prime p such that n is a solution mod p of x^4 = 2, or 0 if no such prime exists.

7, 79, 127, 7, 647, 2399, 23, 937, 4999, 14639, 1481, 28559, 19207, 23, 31, 47, 73, 18617, 79999, 194479, 117127, 5711, 165887, 73, 4663, 113, 233, 707279, 47, 40153, 524287, 191, 167, 257, 439, 267737, 45329, 2313439, 182857, 2825759, 1555847

Offset: 2

Author

Klaus Brockhaus, Nov 28 2001

Keywords

• nonn

Comments

Solutions mod p are represented by integers from 0 to p-1. The following equivalences holds for n > 1: There is a prime p such that n is a solution mod p of x^4 = 2 iff n^4 - 2 has a prime factor > n; n is a solution mod p of x^4 = 2 iff p is a prime factor of n^ 4 - 2 and p > n. n^4 - 2 has at most three prime factors > n, so these factors are the only primes p such that n is a solution mod p of x^4 = 2. The first zero is at n = 1689 (cf. A065903 ). For n such that n^4 - 2 has one resp. two resp. three prime factors > n; cf. A065904 resp. A065905 resp. A065906.

Examples

			a(16) = 31, since 16 is a solution mod 31 of x^4 = 2 and 16 is not a solution mod p of x^4 = 2 for primes p < 31. Although 16^4 = 2 (mod 7), prime 7 is excluded because 7 < 16 and 16 = 2 (mod 7).
		

Crossrefs

Cf. A040098, A065903, A065904, A065905, A065906.

Programs

PARI

a065902(m) = local(n,f,a,j); for(n = 2,m,f = factor(n^4-2); a = matsize(f)[1]; j = 1; while(f[j,1]< = n&&jn,f[j,1],0),","))
a065902(45)

Formula

If n^4 - 2 has prime factors > n, then a(n) = smallest of these prime factors, else a(n) = 0.