In 1770, E. Waring made an assertion these days interpreted as conjecturing that when k is a natural number, all positive integers may be written as the sum of a number g(k) of positive integral kth powers, with g(k) finite. Since the work of Hardy and Littlewood a century ago, attention has largely shifted to the problem of bounding G(k), the least number s having the property that all sufficiently large integers can be written as the sum of s positive integral kth powers. It is known that G(2) = 4 (Lagrange), G(3) ≤ 7 (Linnik), G(4) = 16 (Davenport), and G(5) ≤ 17, G(6) ≤ 24, …, G(20) ≤ 142 (Vaughan and Wooley). For large k one has G(k) ≤ k(log k+log log k+2+o(1)) (Wooley). We report on very recent progress joint with Joerg Bruedern. One or two new world records will be on display.
This video is part of the Number Theory Web Seminar series.
