A385165 -id:A385165 - cp's OEIS frontend

A385219 Multiplicative orders of 2+-i modulo p == 3 (mod 4) that are not divisible by 2 or 3.

7475, 19895, 69445, 87725, 80735, 205975, 504095, 212605, 125081, 1274665, 720055, 181445, 1044005, 492929, 891335, 1346365, 5501795, 7360445, 8179505, 9489095, 10628035, 3850775, 3138905, 14618765, 15377605, 34181, 17907265, 21377825, 23942035, 5047511, 13694965, 6868865, 28713125

Offset: 1

Jianing Song, Jun 22 2025

Primes p == 3 (mod 4) are precisely the rational primes in the ring of Gaussian integers.
Elements in A385165 that are not divisible by 2 or 3.
By definition, a(n) is the multiplicative order of 2+-i modulo A385188(n).

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

Cf. A385165, A385188 (corresponding primes), A385217, A385218.

ord(p) = my(d = divisors((p+1)*znorder(Mod(5, p)))); for(i=1, #d, if(Mod([2, -1; 1, 2], p)^d[i] == 1, return(d[i]))) \\ for a prime p == 3 (mod 4), returns ord(2+-i, p)
forprime(p=3, 1e4, if(p%4==3 && ord(p)%2 && ord(p)%3, print1(ord(p), ", ")))

a(9) = 125081 since it is the multiplicative order of 5 modulo A385188(9) = 5479, and it is divisible by neither 2 nor 3.

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

Original entry on oeis.org

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

Jianing Song, Jun 20 2025

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).

			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).

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

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

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

A384948 Primes p == 3 (mod 4) such that 5 is a primitive root of integers modulo p, but 2+-i are not primitive roots of Gaussian integers modulo p.

Original entry on oeis.org

83, 307, 347, 503, 587, 863, 947, 1103, 1223, 1523, 1567, 1667, 1787, 1907, 2063, 2087, 2267, 2663, 2683, 2687, 2903, 2963, 3167, 3343, 3347, 3623, 3803, 3863, 4283, 4463, 4523, 4643, 4967, 5147, 5303, 5387, 5507, 5563, 5807, 5843, 6047, 6203, 6607, 6863, 6983, 7187, 7247, 7523, 7583

Offset: 1

Jianing Song, Jun 20 2025

For p = A002145(k), A385165(k) divides (p+1) * ord(5,p), since we have (2+-i)^(p+1) == 5 (mod p). Hence if 2+-i are primitive roots of Gaussian integers modulo p, then 5 is a primitive root of integers modulo p. This sequence lists p such that the converse does not hold.

			5 is a primitive root modulo 83, but the multiplicative order of 2+-i modulo 83 in Gaussian integers is not 83^2 - 1 = 6888; it is 2296 = 6888/3. In other words, 2+-i are not generators of (Z[i]/83Z[i])*.

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

By definition, subsequence of A019335, A122870, and A385168.

isprim(p) = my(f = factor(p^2-1)[,1]~); for(i=1, #f, if(Mod([2, -1; 1, 2], p)^((p^2-1)/f[i]) == 1, return(0))); return(1) \\ for a prime p == 3 (mod 4), determines if 2+-i are primitive roots modulo p
isA384948(p) = isprime(p) && (p%4==3) && znorder(Mod(5,p))==p-1 && !isprim(p)

A385190 Primes p == 3 (mod 4), p > 3 such that 1+-i are 24th powers modulo p.

Original entry on oeis.org

31, 127, 191, 223, 383, 479, 863, 1151, 1439, 1471, 1823, 2111, 2143, 2207, 2399, 2591, 2687, 2879, 3167, 3359, 3391, 4127, 4703, 4799, 5087, 5279, 5471, 5503, 6047, 6079, 6143, 6271, 6719, 6911, 7103, 7487, 7583, 8191, 8287, 8447, 8543, 8831, 8863, 9311, 9439, 9631, 9791, 9887

Offset: 1

Jianing Song, Jun 20 2025

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 (1+-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) such that the multiplicative order of 1+-i modulo p (A385163(k)) divides (p^2-1)/24. Since A385165(k) = 4*ord(-4,p), this is also primes p == 3 (mod 4) such that 96*ord(-4,p) divides p^2-1, where ord(a,p) is the multiplicative order of a modulo p.
Sequence is infinite since it contains all primes congruent to 95 modulo 96.
Primes p == 3 (mod 4), p > 3 such that [1,-1;1,1]^((p^2-1)/24) or [1,1;-1,1]^((p^2-1)/24) == I_2 (mod p).
Since 96 divides p^2-1 for p being a term of this sequence, we must have p == 15 (mod 16).

			31 is a term since (1+-i)^((31^2-1)/24) = (-4)^((31^2-1)/96) = 1048576 == 1 (mod 31). Indeed, the solutions to x^24 == 1+i (mod 31) are x == {17-6*i, 16+6*i, 1+8*i, -1+13*i, 9-5*i, 3+5*i} X {+-1, +-i} (mod 31).

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

Cf. A385163, A385191 (2+-i are 24th powers), A002145.

isA385190(p) = isprime(p) && p%16==15 && Mod(-4,p)^((p^2-1)/96) == 1

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A385219 Multiplicative orders of 2+-i modulo p == 3 (mod 4) that are not divisible by 2 or 3.

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

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A384948 Primes p == 3 (mod 4) such that 5 is a primitive root of integers modulo p, but 2+-i are not primitive roots of Gaussian integers modulo p.

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A385190 Primes p == 3 (mod 4), p > 3 such that 1+-i are 24th powers modulo p.

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