Shapely Numbers and Mr. Waring
Malcolm E. Lines · 2020
Edward Waring was a rather modest English mathematician of the eighteenth century whose interest in the connection between shapely numbers and integers led him in 1770 to make the statement that “every positive integer can be made up as the sum of no more than 4 squares, no more than 9 cubes, no more than 19 fourth powers, and so on”. The assertion, named after Edward Waring, was not presented with any accompanying grand theory, but was probably just a plausible surmise on Waring&s;s part arrived at by examining the smaller integers. Nevertheless it is possible to extend Waring&s;s problem for cubes in a number of directions involving other shapely numbers, and even to spaces with more than three dimensions. This chapter concludes with one final Fermat theorem concerning the shapely numbers, again simple to state but as usual not at all easy to prove.