MR WARING'S PROBLE M Jan Tex t Macquarie University, NSW 2109, Australia jan @mpce.mq . edu. a u 1. Introductio n Mr Edward Waring was an English mathematicia n of the 18th century. One of the problems h e considered was this: given a positive integer n, ho w can it be expressed as the sum of integer cubes? I n particular, what is the minimum number of cubes needed? For example, the integer 567 can be expressed trivially as the sum of 567 cubes, namely : 13 + 13 + 13 + . . . + More interestingly, it can be expressed as : 1 3 + 3 3 + 3 3 + 83 or, better still, as : 23 + 63 + 73 Simple investigation will verify that it cannot b e formed from one or two cubes . The minimu m number of cubes for 567 is therefore three . In 1770, Mr Waring asserted that `ever y positive integer can be made up as the sum of n o more than 9 cubes' . By considering the case n 23, it is easily shown that nine cubes ar e occasionally necessary . However, it
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Mr Waring's problem
Hext, Jan
Follow ACM SIGCSE Bulletin
, Volume 26 (4)
Association for Computing Machinery – Dec 1, 1994
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