A097955 Primes p such that p divides 5^((p-1)/2) - 2^((p-1)/2).

3, 13, 31, 37, 41, 43, 53, 67, 71, 79, 83, 89, 107, 151, 157, 163, 173, 191, 197, 199, 227, 239, 241, 271, 277, 281, 283, 293, 307, 311, 317, 347, 359, 373, 397, 401, 409, 431, 439, 443, 449, 467, 479, 521, 523, 547, 557, 563, 569, 587, 599, 601, 613, 631, 641

Offset: 1

Author

Cino Hilliard, Sep 06 2004

Keywords

• nonn
• easy

Comments

Also 3 and primes p such that (p^2 - 1)/24 mod 10 = {0, 7}. - Richard R. Forberg, Aug 31 2013

Also primes p such that x^2 = 10 mod p has integer solutions, or Legendre(10, p) = 1. However, p could be irreducible but not prime in Z[sqrt(10)], especially if p = 3 or 7 mod 10. - Alonso del Arte, Dec 27 2015

Rational primes that decompose in the field Q(sqrt(10)). - N. J. A. Sloane, Dec 26 2017

From Jianing Song, Oct 13 2022: (Start)

Primes p such that kronecker(10,p) = 1 (or equivalently, kronecker(40,p) = 1).

Primes congruent to 1, 3, 9, 13, 27, 31, 37, 39 modulo 40. (End)

Examples

			For p = 13, 5^6 - 2^6 = 15561 is divisible by 13, so 13 is in the sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index to sequences related to decomposition of primes in quadratic fields

Crossrefs

A038879, the sequence of primes that do not remain inert in the field Q(sqrt(10)), is essentially the same.

Cf. A038880 (rational primes that remain inert in the field Q(sqrt(10))).

Programs

Maple

select(p -> isprime(p) and  10 &^ ((p-1)/2)  mod p = 1, [seq(i,i=3..1000,2)]); # Robert Israel, Dec 28 2015

Mathematica

Select[Prime[Range[100]], JacobiSymbol[10, #] == 1 &] (* Alonso del Arte, Dec 27 2015 *)

PARI

\\ s = +-1,d=diff
ptopm1d2(n,x,d,s) = { forprime(p=3,n,p2=(p-1)/2; y=x^p2 + s*(x-d)^p2; if(y%p==0,print1(p","))) }
ptopm1d2(1000, 5, 3, -1)

PARI

isA097955(p) == isprime(p) && kronecker(10,p) == 1 \\ Jianing Song, Oct 13 2022