A319882 Number of ordered pairs (i, j) with 0 < i < j < prime(n)/2 such that (i^4 mod prime(n)) > (j^4 mod prime(n)).

0, 0, 0, 3, 3, 10, 16, 21, 33, 54, 82, 85, 103, 125, 138, 165, 157, 204, 267, 259, 359, 422, 471, 504, 584, 564, 627, 713, 628, 1053, 960, 1213, 1017, 1278, 1275, 1367, 1522, 1671, 1661, 2118, 2038, 2005, 2242, 2330, 2234, 2418, 3194, 3112, 3126

Offset: 2

Author

Zhi-Wei Sun, Sep 30 2018

Keywords

• nonn

Comments

Conjecture: Let p be any odd prime, and let t(p) be the number of ordered pairs (i,j) with 0 < i < j < p/2 and (i^4 mod p) > (j^4 mod p). If p is not congruent to 7 modulo 8, then t(p) == floor((p-1)/8) (mod 2). When p == 7 (mod 8), we have t(p) == (p+1)/8 + (h(-p)+1)/2 (mod 2), where h(-p) denotes the class number of the imaginary quadratic field Q(sqrt(-p)).

See also A319311, A319480 and A319894 for similar conjectures.

Examples

			a(5) = 3 since prime(5) = 11, and the only ordered pairs (i, j) with 0 < i < j < 11/2 and (i^4 mod 11) > (j^4 mod 11) are (2, 3), (2, 4), (3, 4).
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 2..1000
• Zhi-Wei Sun, Quadratic residues and related permutations, arXiv:1809.07766 [math.NT], 2018.

Crossrefs

Cf. A000040, A000583, A319311, A319480, A319894.

Programs

Mathematica

f[k_, p_] := f[k, p] = PowerMod[k, 4, p]; Inv[p_] := Inv[p] = Sum[Boole[f[i, p] > f[j, p]], {j, 2, (p - 1)/2}, {i, 1, j - 1}]; Table[Inv[Prime[n]], {n, 2, 50}]