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The solvability of the equation n = x 2 + y 2 + 6pz 2 (p is a fixed large prime) is proved under some natural congruential conditions and the assumption nm 12 > p 21. As an implication, the solvability of the equation n = x 2 + y 2 + u 3 + v 3 + z 4 + w 16 + t 4k+1 for all sufficiently large n is established. Bibliography: 13 titles.

A bound for the representability of large numbers by ternary quadratic forms and nonhomogeneous waring equations