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

Author

Jianing Song, Jun 20 2025

Keywords

• nonn
• easy

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).

Examples

			1759 is a term since (2+-i)^((1759^2-1)/24) = (-4)^((31^2-1)/96) = 1048576 == 1 (mod 31). 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).
		

Links

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

Crossrefs

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

Programs

PARI

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