It has been shown by Dickson (1) that if (i, j)8 is the number of solutions of

(mod p),

then 64(i,j)8 is expressible for each i,j, as a linear combination with integer coefficients of p, x, y, a, and b where

,

and

a ≡ b ≡ 1 (mod 4),

while the sign of y and b depends on the choice of the primitive root g. There are actually four sets of such formulas depending on whether p is of the form 16n + 1 or 16n + 9 and whether 2 is a quartic residue or not.