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A065902 Smallest prime p such that n is a solution mod p of x^4 = 2, or 0 if no such prime exists. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

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.

LINKS

Table of n, a(n) for n=2..42.

FORMULA

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

EXAMPLE

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).

PROG

(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&&j<a, j++); print1(if(f[j, 1]>n, f[j, 1], 0), ", ")) a065902(45)

CROSSREFS

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

Sequence in context: A261181 A106107 A020471 * A140613 A139945 A023285

Adjacent sequences:  A065899 A065900 A065901 * A065903 A065904 A065905

KEYWORD

nonn

AUTHOR

Klaus Brockhaus, Nov 28 2001

STATUS

approved

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Last modified May 12 07:28 EDT 2021. Contains 343821 sequences. (Running on oeis4.)