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In this paper we prove that certain products and sums of powers of binomial coefficients modulo p = qf + 1, q = a1 + b2, are determined by the parameters x occurring in distinct solutions of the quaternary quadratic partition 16p° = x2 + 2qu2 + 2qv2 + qw2, (x, u, v,w, p) = 1, xw = av2 2buv au2, x ■ 4 (mod q), a > 1. The number of distinct solutions of this partition depends heavily on the class number of the imaginary cyclic quartic field K= Q(i]¡2q+ 2ajq), as well as on the number of roots of unity in K and on the way that p splits into prime ideals in the ring of integers of the field Q(e2,"p/q). Let the four cosets of the subgroup A of quartic residues be given by Cj = 2>A,j = 0,1,2,3, and let 'j-Jl'. 7 = 0,1,2,3. t tec. Let sm and s„ denote the smallest and next smallest of the s¡ respectively. We give new, and unexpectedly simple determinations of n*eCil kf\ and n*eC|i+2 kf\, in terms of the parameters x in the above partition of 16/>°, in the complicated case that arises when the class number of K is > 1 and s,„ # s„.

Products and sums of powers of binomial coefficients mod 𝑝 and solutions of certain quaternary Diophantine systems