The representation of numbers as sums of unlike powers

We are concerned with the problem of finding the least s for which every large natural number n admits a representation $n = x\sbsp{2}{2} + x\sbsp{3}{3} + \cdots + x\sbsp{s+1}{s+1}$, where the numbers $x\sb{i}$ are nonnegative integers. K. F. Roth proved in 1948 that one may take s = 50, and this value has subsequently been reduced to s = 17 in a series of papers by K. Thanigasalam, R. C. Vaughan and J. Brudern. We improve this further, showing that s = 14 is admissible. This is accomplished by adapting a new iterative method, developed by Vaughan and T. D. Wooley for use in Waring's problem, for problems involving mixed powers. As with the other papers on the subject, the proof employs the machinery of the Hardy-Littlewood circle method. We also consider the problem of representing integers n in the form $n = x\sbsp{k}{k} + x\sbsp{k+1}{k+1} + \cdots + x\sbsp{k+s-1}{k+s-1}$, and obtain upper bounds on the number of terms required to represent every sufficiently large n in this form both for general k and for the specific case k = 3. The estimate obtained for general k improves an estimate by E. J. Scourfield. It is conjectured that in fact all large n can be written as the sum of a square, a positive cube and a fourth power of integers, and we give some numerical calculations that show that there are still many exceptions greater than 10$\sp{18}$.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2011

Author dc:creator

• Ford, Kevin Barry

• Mathematics

Statement dc:rights

• Copyright 1994 Ford, Kevin Barry

Language dc:language

eng

Identifier

AAI9503187
(UMI)AAI9503187

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/20414