A385218 -id:A385218 - cp's OEIS frontend

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.

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

A385217 Odd multiplicative orders of 2+-i modulo primes p == 3 (mod 4).

Original entry on oeis.org

13695, 40755, 7475, 19895, 43995, 117855, 138075, 13185, 69445, 87725, 308505, 220665, 567645, 80735, 1103355, 1321125, 1386945, 507795, 1594005, 130995, 205975, 2051325, 2092035, 2216565, 2703975, 1368315, 2750685, 504095, 3039345, 212605, 3342405, 125081, 1274665, 3991725, 152205, 4279275

Offset: 1

Jianing Song, Jun 22 2025

Primes p == 3 (mod 4) are precisely the rational primes in the ring of Gaussian integers.
Odd elements in A385165.
By definition, a(n) is the multiplicative order of 2+-i modulo A385169(n).

			a(49) = 635253 since it is the multiplicative order of 5 modulo A385169(49) = 8731, and it is odd.
a(50) = 57447 since it is the multiplicative order of 5 modulo A385169(50) = 8839, and it is odd.
a(51) = 2029953 since it is the multiplicative order of 5 modulo A385169(51) = 9011, and it is odd.

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

Cf. A385165, A385169 (corresponding primes), A385218, A385219.

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, print1(ord(p), ", ")))

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.

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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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A385217 Odd multiplicative orders of 2+-i modulo primes p == 3 (mod 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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