A385188 -id:A385188 - cp's OEIS frontend

A385169 Primes p == 3 (mod 4) such that the multiplicative order of 2+-i modulo p in Gaussian integers (A385165) is odd.

331, 571, 599, 691, 839, 971, 1051, 1171, 1291, 1451, 1571, 1879, 2131, 2411, 2971, 3251, 3331, 3491, 3571, 3691, 3851, 4051, 4091, 4211, 4651, 4679, 4691, 4919, 4931, 5051, 5171, 5479, 5531, 5651, 5839, 5851, 5879, 6011, 6599, 6679, 6691, 7079, 7211, 7331, 7691, 8011, 8039, 8171, 8731, 8839, 9011, 9371, 9811

Offset: 1

Jianing Song, Jun 20 2025

Primes p == 3 (mod 4) are precisely the rational primes in the ring of Gaussian integers.
Let ord(a,m) be the multiplicative order of a modulo m. (Of course if a and m are integers, it doesn't matter if the base ring is Z or Z[i]). For a prime p == 3 (mod 4), we have that ord(2+-i,p) is divisible by ord(5,p), and that ord(2+-i,p) divides (p+1) * ord(5,p). What's more, ord(2+-i,p) divides (p^2-1)/2 if and only if 5 is a quadratic residue of integers modulo p. (See A385165).
As a result, if ord(2+-i,p) is not divisible by 8, then ord(5,p) is odd:
  - Of course this is true if ord(2+-i,p) is odd.
  - If ord(2+-i,p) == 2 (mod 4) and ord(5,p) is even, then ord(2+-i,p)/ord(5,p) is odd, and so ord(2+-i,p) divides ((p+1)/4) * ord(5,p), then ord(5,p) is odd. This implies that ord(2+-i,p) is odd, a contradiction.
  - If ord(2+-i,p) == 4 (mod 8) and ord(5,p) is even (we have ord(5,p) == 2 (mod 4) since p == 3 (mod 4)), then ord(2+-i,p)/ord(5,p) == 2 (mod 4), and so ord(2+-i,p) divides ((p+1)/2) * ord(5,p), then ord(5,p) is odd. This implies that ord(2+-i,p) == 2 (mod 4), a contradiction.
From the above paragraph, this sequence is also primes p == 3 (mod 4) such that ord(2+-i,p)/ord(5,p) is odd.

			8731 is a term since (2+-i)^635253 == 1 (mod 8731), and 635253 is odd.
8839 is a term since (2+-i)^57447 == 1 (mod 8839), and 57447 is odd.
9011 is a term since (2+-i)^2029953 == 1 (mod 9011), and 2029953 is odd.

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

Cf. A385165, A385179, A385192, A385217 (the actual multiplicative orders).

A385188 < this sequence < A385180 < A385167 < intersection of A122869 and A385168, where Ax < Ay means that Ax is a subsequence of Ay.

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)
isA385169(p) = isprime(p) && p%4==3 && ord(p)%2

A385179 Primes p == 3 (mod 4) such that the multiplicative order of 2+-i modulo p in Gaussian integers (A385165) is congruent to 2 modulo 4.

Original entry on oeis.org

11, 131, 211, 251, 491, 811, 919, 1039, 1091, 1319, 1399, 1531, 1811, 1931, 2011, 2251, 2371, 2531, 2731, 2851, 3011, 3079, 3371, 3931, 4079, 4451, 4519, 4759, 5011, 5639, 6091, 6131, 6211, 6359, 6451, 6491, 6571, 6971, 7411, 7451, 7559, 7639, 8291, 8719, 8971, 9091, 9491, 9719, 9839, 9851, 9931

Offset: 1

Jianing Song, Jun 20 2025

Primes p == 3 (mod 4) are precisely the rational primes in the ring of Gaussian integers.
Let ord(a,m) be the multiplicative order of a modulo m. (Of course if a and m are integers, it doesn't matter if the base ring is Z or Z[i]). For a prime p == 3 (mod 4), we have that ord(2+-i,p) is divisible by ord(5,p), and that ord(2+-i,p) divides (p+1) * ord(5,p). What's more, ord(2+-i,p) divides (p^2-1)/2 if and only if 5 is a quadratic residue of integers modulo p. (See A385165).
As a result, if ord(2+-i,p) is not divisible by 8, then ord(5,p) is odd:
  - Of course this is true if ord(2+-i,p) is odd.
  - If ord(2+-i,p) == 2 (mod 4) and ord(5,p) is even, then ord(2+-i,p)/ord(5,p) is odd, and so ord(2+-i,p) divides ((p+1)/4) * ord(5,p), then ord(5,p) is odd. This implies that ord(2+-i,p) is odd, a contradiction.
  - If ord(2+-i,p) == 4 (mod 8) and ord(5,p) is even (we have ord(5,p) == 2 (mod 4) since p == 3 (mod 4)), then ord(2+-i,p)/ord(5,p) == 2 (mod 4), and so ord(2+-i,p) divides ((p+1)/2) * ord(5,p), then ord(5,p) is odd. This implies that ord(2+-i,p) == 2 (mod 4), a contradiction.
From the above paragraph, this sequence is also primes p == 3 (mod 4) such that ord(2+-i,p)/ord(5,p) == 2 (mod 4).

			919 is a term since (2+-i)^21114 == 1 (mod 919), (2+-i)^(21114/2) !== 1 (mod 919), and we have 21114 == 2 (mod 4).

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

Cf. A385165, A385169, A385188, A385218 (the actual multiplicative orders).

Subsequence of A385167, which itself lies in the intersection of A122869 and A385168.

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)
isA385179(p) = isprime(p) && p%4==3 && ord(p)%4==2

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

Original entry on oeis.org

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

cp's OEIS Frontend

A385169 Primes p == 3 (mod 4) such that the multiplicative order of 2+-i modulo p in Gaussian integers (A385165) is odd.

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A385179 Primes p == 3 (mod 4) such that the multiplicative order of 2+-i modulo p in Gaussian integers (A385165) is congruent to 2 modulo 4.

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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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