Redirected from Warings problem
For every k, we denote the least such s by g(k). Note we have g(1) = 1.
Lagrange's Four Square Theorem from 1770 states that every natural number is the sum of at most four squares; since three squares are not enough, this theorem establishes g(2) = 4. Lagrange's Four Square Theorem was conjectured by Fermat in 1640 and was first stated in 1621.
g(3) = 9 was established from 1909 to 1912 by Wieferich[?] and A. J. Kempner, g(4) = 19 in 1986 by R. Balasubramanian, F. Dress, and J.-M. Deshouillers, g(5) = 37 in 1964 by Jing-run Chen and g(6) = 73 in 1940 by Pillai[?]. The values g(3) = 9 and g(4) = 19 had already been conjectured in 1778 by Waring.
All the other values of g are now also known, as a result of work by Dickson, Pillai, Rubugunday and Niven. Their formula contains two cases, and it is conjectured that the second case never occurs; in the first case, the formula reads
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