A097959 Primes p such that p divides 6^((p-1)/2) - 5^((p-1)/2).

7, 13, 17, 19, 29, 37, 71, 83, 101, 103, 107, 113, 127, 137, 139, 149, 157, 191, 211, 223, 227, 233, 239, 241, 257, 269, 277, 311, 331, 347, 353, 359, 367, 373, 379, 389, 397, 409, 431, 443, 461, 463, 467, 479, 487, 499, 509, 563, 571, 587, 593, 599, 601, 607

Offset: 1

Author

Cino Hilliard, Sep 06 2004

Keywords

• nonn
• easy

Comments

From Jianing Song, Oct 13 2022: (Start)

Rational primes that decompose in the field Q(sqrt(30)).

Primes p such that kronecker(30,p) = 1 (or equivalently, kronecker(120,p) = 1).

Primes congruent to 1, 7, 13, 17, 19, 29, 37, 49, 71, 83, 91, 101, 103, 107, 113, 119 modulo 120. (End)

Examples

			7 is a term since it is a prime and 6^((7-1)/2) - 5^((7-1)/2) = 6^3 - 5^3 = 91 = 7*13 is divisible by 7.
		

Links

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

Crossrefs

A038903, the sequence of primes that do not remain inert in the field Q(sqrt(30)), is essentially the same.

Cf. A038904 (rational primes that remain inert in the field Q(sqrt(30))).

Programs

Mathematica

Select[Prime[Range[200]],Divisible[6^((#-1)/2)-5^((#-1)/2),#]&] (* Harvey P. Dale, Jun 06 2018 *)
Select[Range[3, 600, 2], PrimeQ[#] && PowerMod[5, (# - 1)/2, #] == PowerMod[6, (# - 1)/2, #] &] (* Amiram Eldar, Apr 07 2021 *)

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, 6, 1, -1)

PARI

isA097959(p) == isprime(p) && kronecker(30, p) == 1 \\ Jianing Song, Oct 13 2022

Extensions

Definition clarified by Harvey P. Dale, Jun 06 2018