Applications of the Hardy-Littlewood Circle Method

1 Waring’s Problem Problem 1. (Waring’s) For every natural number k ≥ 2 there exists a positive integer s such that every natural number is the sum of at most s k powers of natural number (for example, every natural number is the sum of at most 4 squares, or 9 cubes, or 19 fourth powers, etc.). The affirmative answer, known as the Hilbert-Waring theorem, was provided by Hilbert in 1909. Let A = (am) denote a strictly increasing sequences of nonnegative integers. Consider F (z) = ∞ ∑