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2016 Every integer greater than 454 is the sum of at most seven positive cubes
Samir Siksek
Algebra Number Theory 10(10): 2093-2119 (2016). DOI: 10.2140/ant.2016.10.2093

Abstract

A long-standing conjecture states that every positive integer other than

$\begin{array}{cc}\phantom{\rule{0.3em}{0ex}}15,\phantom{\rule{2.77626pt}{0ex}}22,\phantom{\rule{2.77626pt}{0ex}}23,\phantom{\rule{2.77626pt}{0ex}}50,\phantom{\rule{2.77626pt}{0ex}}114,\phantom{\rule{2.77626pt}{0ex}}167,\phantom{\rule{2.77626pt}{0ex}}175,\phantom{\rule{2.77626pt}{0ex}}186,\phantom{\rule{2.77626pt}{0ex}}212,& \\ 231,\phantom{\rule{2.77626pt}{0ex}}238,\phantom{\rule{2.77626pt}{0ex}}239,\phantom{\rule{2.77626pt}{0ex}}303,\phantom{\rule{2.77626pt}{0ex}}364,\phantom{\rule{2.77626pt}{0ex}}420,\phantom{\rule{2.77626pt}{0ex}}428,\phantom{\rule{2.77626pt}{0ex}}454& \end{array}$

is a sum of at most seven positive cubes. This was first observed by Jacobi in 1851 on the basis of extensive calculations performed by the famous computationalist Zacharias Dase. We complete the proof of this conjecture, building on previous work of Linnik, Watson, McCurley, Ramaré, Boklan, Elkies, and many others.

Citation

Samir Siksek. "Every integer greater than 454 is the sum of at most seven positive cubes." Algebra Number Theory 10 (10) 2093 - 2119, 2016. https://doi.org/10.2140/ant.2016.10.2093

Information

Received: 6 January 2016; Revised: 21 August 2016; Accepted: 23 September 2016; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 06664746
MathSciNet: MR3582015
Digital Object Identifier: 10.2140/ant.2016.10.2093

Subjects:
Primary: 11P05

Keywords: cubes , sums of cubes , Waring