Some Famous Problems of the Theory of Numbers and in particular Waring' Problem: An Inaugural Lecture delivered before the University of Oxford

Abstract

THE theory of the integral numbers is a subject in which it is frequently easy to conjecture new results and extremely difficult to prove them. An example of a result which must have been based on conjecture is known as Waring's theorem, that every positive integer is the sum of nine (or fewer) positive cubes, of nineteen (or fewer) biquadrates, and so on. A proof of this result, asserted in 1782, was first approached by Prof. Hubert, of Göttingen, who showed in 1909 that every integer n is the sum of a finite number not exceeding g(k), independent of n, of exact kth powers. It has been established, by transcendental analysis developed long since the days of Waring, that g(3) = 9 as asserted by him, but whether g(4) = 19 is still uncertain, though this number has been shown not to exceed 37. The only positive integers known to be inexpressible as a sum of eight cubes (at most) are 23 and 239.

Some Famous Problems of the Theory of Numbers and in particular Waring's Problem: An Inaugural Lecture delivered before the University of Oxford.

By Prof. G. H. Hardy. Pp. 34. (Oxford: At the Clarendon Press, 1920.) Price 1s. 6d. net.