A385191 Primes p == 3 (mod 4), p > 3 such that 2+-i are 24th powers modulo p.

599, 691, 1039, 1291, 1451, 1759, 2411, 2879, 3079, 3491, 3851, 4519, 4639, 4919, 5051, 5479, 5519, 5531, 5639, 5879, 6011, 6079, 6599, 6719, 7079, 7691, 8011, 8039, 8171, 8731, 9439, 9839, 10799, 11159, 11239, 11411, 11491, 12239, 12799, 13291, 13679, 13759, 13879, 14011, 14639

Offset

1,1

Comments

Note that the primes congruent to 3 modulo 4 are precisely the rational primes in the ring of Gaussian integers.

Primes p == 3 (mod 4), p > 3 such that (2+-i)^((p^2-1)/24) == 1 (mod p). Note that p^2-1 is always divisible by 24 for primes p > 3.

Primes p = A002145(k) > 3 such that the multiplicative order of 2+-i modulo p (A385165(k)) divides (p^2-1)/24.

Primes p == 3 (mod 4), p > 3 such that [2,-1;1,2]^((p^2-1)/24) or [2,1;-1,2]^((p^2-1)/24) == I_2 (mod p).

Note that if (x+-y*i)^24 == 1+-i (mod p) for some integers x, y, then (x^2+y^2)^24 == 5 (mod p), so 5 must be a quadratic residue (in rational integers) modulo p. By definition, we have p == 11, 19 (mod 20).

Links

Jianing Song, Table of n, a(n) for n = 1..10000

Example

1759 is a term since (2+-i)^((1759^2-1)/24) == 1 (mod 1759). Indeed, the solutions to x^24 == 2+i (mod 1759) are x == {441+580i, -43+860i, -292+683i, -251+779i, -635+872i, 736-648i} X {+-1, +-i} (mod 1759). [Typo corrected by Jianing Song, Mar 15 2026]

Prog

(PARI) isA385191(p) = p>3 && isprime(p) && p%4==3 && Mod([2, -1; 1, 2], p)^((p^2-1)/24) == 1

Crossrefs

Cf. A385165, A385190 (1+-i are 24th powers), A002145, A122869. A385188 is a subsequence.

Sequence in context: A135846 A035209 A356531 * A385188 A135847 A106762

Adjacent sequences: A385188 A385189 A385190 * A385192 A385193 A385194

Keyword

nonn,easy

Author

Jianing Song, Jun 20 2025

Status

approved