In this paper, we determine all triplets of positive integers a, b, and c such that every nonnegative integer can be represented as fa,b c (x, y, z, w) = ax2 + by2 + c(z2 + zw + w2) with x, y, z, w 2 Z. Furthermore, we prove that fa,b c can represent all the nonnegative integers if it represents 1, 2, 3, 5, 6, and 10.
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