THE CONGRUENCE ax1x2 ··· xk + bxk+1xk+2 ··· x2k ≡ c (mod p)

For prime p and integers a, b, c with p ∤ b, we obtain solutions of the congruence a x 1 x 2 ⋯ x k + b x k + 1 x k + 2 ⋯ x 2 k ≡ c ( mod p ) in a cube B with edge length B. For a cube in general position, we show that if p ∤ abc and k ≥ 5, then the congruence above has a solution in any cube with edge length B ≫ p 1 4 + 1 2 ( k + 1.95 ) + ε . Estimates are given for the case p|c as well, and improvements are given for small k. For cubes cornered at the origin, 1 ≤ xi ≤ B for all i, we obtain a solution provided only that B ≫ p 3 2 k + O ( k log log p ) . Under the assumption of GRH best possible estimates are given. Boxes with unequal edge lengths are also discussed.