A385224 Primes p such that multiplicative order of -4 modulo p is odd.

5, 13, 29, 37, 41, 53, 61, 101, 109, 113, 137, 149, 157, 173, 181, 197, 229, 269, 277, 293, 313, 317, 349, 373, 389, 397, 409, 421, 457, 461, 509, 521, 541, 557, 569, 593, 613, 653, 661, 677, 701, 709, 733, 757, 761, 773, 797, 809, 821, 829, 853, 857, 877, 941, 953, 997

Offset: 1

Author

Jianing Song, Jun 22 2025

Keywords

• nonn
• easy

Comments

The multiplicative order of -4 modulo a(n) is A385230(n).

Different from A133204: 593 is here but not in A133204, and 1601 is in A133204 but not here.

The sequence contains no primes congruent to 3 modulo 4 and all primes congruent to 5 modulo 8:

- If p is a term of this sequence, then -4 is a quadratic residue modulo p, so p == 1 (mod 4);

- For p == 1 (mod 4), we have (-4)^((p-1)/4) == (+-1+-i)^(p-1) == 1 (mod p), where i is a solution to i^2 == -1 (mod p).

Conjecture: this sequence has density 1/3 among the primes.

Links

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

Crossrefs

Subsequence of A002144 (primes congruent to 1 modulo 4).

Contains A007521 (primes congruent to 5 or modulo 8) as a proper subsequence.

Cf. A385230 (the actual multiplicative orders).

Cf. other bases: A014663 (base 2), A385220 (base 3), A385221 (base 4), A385192 (base 5), A163183 (base -2), A385223 (base -3), this sequence (base -4), A385225 (base -5).

Cf. A133204.

Programs

Mathematica

Select[Prime[Range[200]], OddQ[MultiplicativeOrder[-4, #]] &] (* Paolo Xausa, Jun 28 2025 *)

PARI

isA385224(p) = isprime(p) && (p!=2) && znorder(Mod(-4,p))%2