Define $g(k) = \min \{ s :$ every positive integer can be written as a sum of $k$th powers of natural numbers with atmost $s$ summands$\}$. Lagrange proved that $g(2) = 4$. Waring conjectured that $g(3) = 9, g(4) = 19$ and so on.

In fact, in this question, there has been a lot of contribution from Indian mathematicians. The method of attacking this problem is called the circle method and it originates from a seminal paper of Hardy and Ramanujan. The final result owes a lot to the contributions of S.S. Pillai. The analogous question over number fields was settled by C.P. Ramanujam. We shall explain their contributions toward this problem.

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Last updated: 20 Apr 2024