A385166 -id:A385166 - cp's OEIS frontend

A385165 Let p = A002145(n) be the n-th prime == 3 (mod 4); a(n) is the multiplicative order of 2+-i modulo p in Gaussian integers.

8, 48, 30, 180, 528, 96, 1848, 2208, 1740, 1496, 360, 1560, 2296, 10608, 11448, 5376, 4290, 1932, 11400, 8856, 27888, 16020, 1216, 3300, 3710, 49728, 51528, 14280, 3150, 69168, 7344, 80088, 8568, 48360, 13695, 40136, 6444, 44896, 7980, 146688, 29260, 92880, 48180

Offset: 1

Jianing Song, Jun 20 2025

A002145 are precisely the rational primes in the ring of Gaussian integers.
From the representation of complex numbers as 2 X 2 matrices, a(n) is also the multiplicative order of the matrix [2,-1;1,2] or [2,1;-1,2] modulo p.
a(n) is divisible by ord(5,p): If (2+-i)^n == 1 (mod p), then 5^n == 1 (mod p).
a(n) divides (p+1) * ord(5,p), since we have (2+-i)^(p+1) == 5 (mod p).
If 5 is a quadratic residue modulo p, then ord(5,p) divides (p-1)/2, and so a(n) divides (p^2-1)/2. Conversely, if a(n) divides (p^2-1)/2, then (x+-y*i)^2 == 2+-i (mod p) for some integers x, y, and so (x^2+y^2)^2 == 5 (mod p), which means that 5 is a quadratic residue modulo p.

			The multiplicative order of 2+-i modulo A002145(3) = 11 is a(3) = 30, since (2+-i)^30 == 1 (mod 11), and 30 is the smallest such exponent.

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

Cf. A002145, A211241, A385163 (multiplicative order of 1+-i), A385166.

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, 1e3, if(p%4==3, print1(ord(p), ", ")))

A385168 Primes p == 3 (mod 4) such that (p+1) * ord(5,p) / ord(2+-i,p) > 1. Here ord(a,m) is the multiplicative order of a modulo m.

Original entry on oeis.org

11, 79, 83, 131, 139, 191, 199, 211, 239, 251, 307, 331, 347, 359, 419, 439, 479, 491, 503, 571, 587, 599, 659, 691, 719, 811, 839, 863, 919, 947, 971, 1019, 1039, 1051, 1091, 1103, 1171, 1223, 1231, 1279, 1291, 1319, 1399, 1439, 1451, 1499, 1523, 1531, 1559, 1567, 1571, 1619, 1667, 1759

Offset: 1

Jianing Song, Jun 20 2025

Of course if a and m are integers, it doesn't matter if the base ring is Z or Z[i] for ord(a,m).
List of p = A002145(k) such that A385166(k) > 1.
The smallest terms congruent to 1 or 4 modulo 5 that are not in A385167 are 139, 191, 419, 659, ...
The smallest terms congruent to 2 or 3 modulo 5 that are not in A384948 are 5683, 6287, 9463, 9923, ...

Jianing Song, Table of n, a(n) for n = 1..10001

Cf. A002145, A385165 (list of ord(2+-i,p)), A385166 (list of (p+1) * ord(5,p) / ord(2+-i,p)).

A384948 and A385167 (which contains A385180) are subsequences.

quot(p) = my(z = znorder(Mod(5,p)), d = divisors((p+1)*z)); for(i=1, #d, if(Mod([2,-1;1,2],p)^d[i] == 1, return((p+1)*z/d[i]))) \\ for a prime p == 3 (mod 4), returns (p+1) * ord(5,p) / ord(2+-i, p)
isA385168(p) = isprime(p) && p%4==3 && quot(p) > 1

139 is a term since the multiplicative order of 2+-i modulo 139 is 1932, and (140*ord(5,139))/1932 = 5 > 1.

5683 is a term since the multiplicative order of 2+-i modulo 5683 is 1537928, and (5684*ord(5,5683))/1537928 = 7 > 1.

A385167 Primes p == 3 (mod 4) such that (p+1) * ord(5,p) / ord(2+-i,p) is even. Here ord(a,m) is the multiplicative order of a modulo m.

Original entry on oeis.org

11, 79, 131, 199, 211, 239, 251, 331, 359, 439, 479, 491, 571, 599, 691, 719, 811, 839, 919, 971, 1039, 1051, 1091, 1171, 1279, 1291, 1319, 1399, 1439, 1451, 1531, 1559, 1571, 1759, 1811, 1879, 1931, 1999, 2011, 2039, 2131, 2239, 2251, 2371, 2399, 2411, 2531, 2719, 2731, 2851, 2879, 2971, 2999

Offset: 1

Jianing Song, Jun 20 2025

Of course if a and m are integers, it doesn't matter if the base ring is Z or Z[i] for ord(a,m).
List of p = A002145(k) such that A385166(k) is even.
Since in this case d(p) divides (p^2-1)/2, 5 must be a quadratic residue modulo p (see A385165).

			359 is a term since the multiplicative order of 2+-i modulo 359 is 6444, and (360*ord(5,359))/6444 = 10 is even.

Jianing Song, Table of n, a(n) for n = 1..10001

Cf. A002145, A385165 (list of ord(2+-i,p)), A385166 (list of (p+1) * ord(5,p) / ord(2+-i,p)).

Subsequence of the intersection of A122869 and A385168. Contains A385180 as a subsequence.

quot(p) = my(z = znorder(Mod(5,p)), d = divisors((p+1)*z)); for(i=1, #d, if(Mod([2,-1;1,2],p)^d[i] == 1, return((p+1)*z/d[i]))) \\ for a prime p == 3 (mod 4), returns (p+1) * ord(5,p) / ord(2+-i, p)
isA385167(p) = isprime(p) && p%4==3 && quot(p)%2==0

A385180 Primes p == 3 (mod 4) such that (p+1) * ord(5,p) / ord(2+-i,p) is divisible by 4. Here ord(a,m) is the multiplicative order of a modulo m.

Original entry on oeis.org

331, 571, 599, 691, 839, 919, 971, 1039, 1051, 1171, 1279, 1291, 1319, 1399, 1439, 1451, 1571, 1759, 1879, 2131, 2411, 2879, 2971, 3079, 3251, 3331, 3491, 3571, 3691, 3851, 4051, 4079, 4091, 4211, 4519, 4639, 4651, 4679, 4691, 4759, 4919, 4931, 5051, 5119, 5171, 5279, 5479, 5519, 5531

Offset: 1

Jianing Song, Jun 20 2025

Of course if a and m are integers, it doesn't matter if the base ring is Z or Z[i] for ord(a,m).
List of p = A002145(k) such that A385166(k) is divisible by 4.
Since in this case d(p) divides (p^2-1)/2, 5 must be a quadratic residue modulo p (see A385165).
By definition, a term that is in neither A385169 nor A385179 must be congruent to 31 or 79 modulo 80. The smallest such term is p = 1759 (ord(2+-i,p) = ((p+1)/4) * ord(5,p) = 128920); even if 1039 == 79 (mod 80), we have ord(2+-i,p) =  ((p+1)/8) * ord(5,p) = 22490 == 2 (mod 4), which means that 1039 is in A385179.

			571 is a term since the multiplicative order of 2+-i modulo 571 is 40755, and (572*ord(5,571))/40755 = 4 is divisible by 4.

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

Cf. A002145, A385165 (list of ord(2+-i,p)), A385166 (list of (p+1) * ord(5,p) / ord(2+-i,p)).

Subsequence of A385167, which is itself a subsequence of intersection of A122869 and A385168.

quot(p) = my(z = znorder(Mod(5,p)), d = divisors((p+1)*z)); for(i=1, #d, if(Mod([2,-1;1,2],p)^d[i] == 1, return((p+1)*z/d[i]))) \\ for a prime p == 3 (mod 4), returns (p+1) * ord(5,p) / ord(2+-i, p)
isA385180(p) = isprime(p) && p%4==3 && quot(p)%4==0

cp's OEIS Frontend

A385165 Let p = A002145(n) be the n-th prime == 3 (mod 4); a(n) is the multiplicative order of 2+-i modulo p in Gaussian integers.

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A385168 Primes p == 3 (mod 4) such that (p+1) * ord(5,p) / ord(2+-i,p) > 1. Here ord(a,m) is the multiplicative order of a modulo m.

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A385167 Primes p == 3 (mod 4) such that (p+1) * ord(5,p) / ord(2+-i,p) is even. Here ord(a,m) is the multiplicative order of a modulo m.

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A385180 Primes p == 3 (mod 4) such that (p+1) * ord(5,p) / ord(2+-i,p) is divisible by 4. Here ord(a,m) is the multiplicative order of a modulo m.

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