cp's OEIS Frontend

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

%I A065902 #7 Feb 10 2025 15:14:55
%S A065902 7,79,127,7,647,2399,23,937,4999,14639,1481,28559,19207,23,31,47,73,
%T A065902 18617,79999,194479,117127,5711,165887,73,4663,113,233,707279,47,
%U A065902 40153,524287,191,167,257,439,267737,45329,2313439,182857,2825759,1555847
%N A065902 Smallest prime p such that n is a solution mod p of x^4 = 2, or 0 if no such prime exists.
%C A065902 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.
%F A065902 If n^4 - 2 has prime factors > n, then a(n) = smallest of these prime factors, else a(n) = 0.
%e A065902 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).
%o A065902 (PARI)
%o A065902 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&&j<a,j++); print1(if(f[j,1]>n,f[j,1],0),","))
%o A065902 a065902(45)
%Y A065902 Cf. A040098, A065903, A065904, A065905, A065906.
%K A065902 nonn
%O A065902 2,1
%A A065902 _Klaus Brockhaus_, Nov 28 2001