The representation of numbers as sums of unlike powers
Ford, Kevin Barry
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https://hdl.handle.net/2142/20414
Description
Title
The representation of numbers as sums of unlike powers
Author(s)
Ford, Kevin Barry
Issue Date
1994
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
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$\sp{18}$.
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