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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)).
4

%I #25 Oct 01 2018 03:51:24

%S 0,0,0,3,3,10,16,21,33,54,82,85,103,125,138,165,157,204,267,259,359,

%T 422,471,504,584,564,627,713,628,1053,960,1213,1017,1278,1275,1367,

%U 1522,1671,1661,2118,2038,2005,2242,2330,2234,2418,3194,3112,3126

%N 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)).

%C 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)).

%C See also A319311, A319480 and A319894 for similar conjectures.

%H Zhi-Wei Sun, <a href="/A319882/b319882.txt">Table of n, a(n) for n = 2..1000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1809.07766">Quadratic residues and related permutations</a>, arXiv:1809.07766 [math.NT], 2018.

%e 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).

%t 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}]

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

%K nonn

%O 2,4

%A _Zhi-Wei Sun_, Sep 30 2018