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A097955
Primes p such that p divides 5^((p-1)/2) - 2^((p-1)/2).
4
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
1,1
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)
EXAMPLE
For p = 13, 5^6 - 2^6 = 15561 is divisible by 13, so 13 is in the sequence.
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 *)
PROG
(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
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))).
Sequence in context: A296014 A273337 A273769 * A320587 A077717 A235265
KEYWORD
nonn,easy
AUTHOR
Cino Hilliard, Sep 06 2004
STATUS
approved