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

1,1

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)

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Index to sequences related to decomposition of primes in quadratic fields

Example

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.

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 *)

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, 6, 1, -1)
(PARI) isA097959(p) == isprime(p) && kronecker(30, p) == 1 \\ Jianing Song, Oct 13 2022

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))).

Sequence in context: A079698 A237609 A038906 * A156543 A090863 A045979

Adjacent sequences: A097956 A097957 A097958 * A097960 A097961 A097962

Keyword

nonn,easy

Author

Cino Hilliard, Sep 06 2004

Extensions

Definition clarified by Harvey P. Dale, Jun 06 2018

Status

approved