#### Vol. 10, No. 10, 2016

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Every integer greater than 454 is the sum of at most seven positive cubes

### Samir Siksek

Vol. 10 (2016), No. 10, 2093–2119
##### 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.

##### Keywords
Waring, cubes, sums of cubes
Primary: 11P05
##### Supplementary material

Magma scripts for the verification of the computations described in this paper, with their output (260 Mbytes)

##### Milestones
Received: 6 January 2016
Revised: 21 August 2016
Accepted: 23 September 2016
Published: 9 December 2016
##### Authors
 Samir Siksek Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom