# Sum of four cubes problem

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Sum_of_four_cubes_problem
> Markdown URL: https://mediated.wiki/source/Sum_of_four_cubes_problem.md
> Source: https://en.wikipedia.org/wiki/Sum_of_four_cubes_problem
> Source revision: 1333139585
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{Short description|Asks whether each integer is a sum of four cubes}}

{{unsolved|mathematics|Is every integer the sum of four perfect cubes?}}
The '''sum of four cubes problem'''<ref>Referred to as the "four cube problem" in H. Davenport, ''The Higher Arithmetic: An Introduction to the Theory of Numbers'', Cambridge University Press, 7th edition, 1999, p. 173, 177.</ref> asks whether every [integer](/source/integer) is the [sum](/source/Summation) of four [cubes](/source/Cube_(algebra)) of integers.  It is conjectured the answer is affirmative,<ref name="Sierpiński and Schinzel, 1959">W. Sierpiński, A. Schinzel. Sur les sommes de quatre cubes. ''Acta Arithmetica'', v. 4, No. 1, 1959, available online at [https://bibliotekanauki.pl/articles/1395061.pdf].</ref> but this conjecture has been neither proven nor disproven, at least as reported in 1982 by Philippe Revoy<ref name="Revoy 1982">Philippe Revoy, “Sur les sommes de quatre cubes”, ''L’Enseignement Mathématique'', t. 29, 1983, p. 209-220, online [http://retro.seals.ch/digbib/view2?pid=ens-001:1983:29::348 here] or [https://www.e-periodica.ch/cntmng?type=pdf&pid=ens-001:1983:29::83 here], p. 209.</ref> and again in 2004 by [Henri Cohen](/source/Henri_Cohen_(number_theorist)).<ref name="Cohen 2004">[Henri Cohen](/source/Henri_Cohen_(number_theorist)), 2004, available via [https://web.archive.org/web/20120118133909/http://www.math.u-bordeaux1.fr/~cohen/sum4cub.ps|Internet Archive] of the paper from Cohen's website.</ref> Some of the cubes may be [negative number](/source/negative_number)s, in contrast to [Waring's problem](/source/Waring's_problem) on sums of cubes, where they are required to be positive.

== Partial results==

This question has been answered for some categories of integers, proving in each case that every integer in the category can be expressed as the sum of 4 cubes of integers.  There are no known categories or specific numbers where this has been proven impossible.<ref name="Revoy 1982"/><ref name="Cohen 2004"/>

=== Proof for most integers ===

In 1959, [W. Sierpiński](/source/Wac%C5%82aw_Sierpi%C5%84ski) and A. Schinzel proved that all integers in the following categories may be expressed as the sum of 4 cubes of integers:<ref name="Sierpiński and Schinzel, 1959"/>
* integers [congruent](/source/Modular_arithmetic) to 0 [modulo](/source/Modular_arithmetic) 6
* integers congruent to 3 modulo 6
* integers congruent to 1 modulo 18
* integers congruent to 7 modulo 18
* integers congruent to 8 modulo 18

Sierpiński and Schinzel also provide an explanation of how the identities used in the proofs of these categories can be used to derive a complementary identity by replacing <math>x</math> with <math>-x</math> throughout the identity, which proves the [opposite](/source/Additive_inverse) category as the original identity.<ref name="Sierpiński and Schinzel, 1959"/>

Taken together, this proved all integers except those congruent to ±2, ±4, and ±5 modulo 18 can be expressed as the sum of 4 cubes.<ref name="Sierpiński and Schinzel, 1959"/>

In addition, Sierpiński and Schinzel provided tables showing solutions for many specific integers in the range of 2 to 300, stating that these tables (in addition to the identities and methods earlier in the paper) together prove that all integers with [absolute value](/source/absolute_value) less than or equal to 300, with the exceptions of ±148, ±257, and ±284, may be decomposed into the sum of 4 cubes.<ref name="Sierpiński and Schinzel, 1959"/>

In 2004, [Henri Cohen](/source/Henri_Cohen_(number_theorist)) provided a simpler summarization of these results, via the following identities (some of which are nearly identical to those used by Sierpiński and Schinzel):<ref name="Cohen 2004"/>

<math display=block>\begin{align}
6x &= (x+1)^{3}+(x-1)^{3}-x^{3}-x^{3} \\
6x+3 &= x^3+(-x+4)^3+(2x-5)^3+(-2x+4)^3 \\
18x+1 &= (2x+14)^3+(-2x-23)^3+(-3x-26)^3+(3x+30)^3 \\
18x+7 &= (x+2)^3+(6x-1)^3+(8x-2)^3+(-9x+2)^3 \\
18x+8 &= (x-5)^3+(-x+14)^3+(-3x+29)^3+(3x-30)^3\ .
\end{align}</math>

=== 18x±2 case ===

In 1966, {{ill|V. A. Demjanenko|de|Wadim Andrejewitsch Demjanenko}} provided the following identities:<ref name="Demjanenko 1966">V.A. Demjanenko, "On sums of four cubes", Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, vol. 54, no. 5, 1966, p. 63-69, available online at [http://www.mathnet.ru/links/59d5c31718f04085ac309f75b4472007/ivm2756.pdf the site Math-Net.Ru].</ref>

<math display=block>\begin{align}
54x + 2 & = (29484x^{2} + 2211x + 43)^{3} + (-29484x^{2} - 2157x - 41)^{3} + (9828x^{2} + 485x + 4)^{3} + (-9828x^{2} - 971x - 22)^{3} \\
54x + 20 & = (3x - 11)^{3} + (-3x + 10)^{3} + (x + 2)^{3} + (-x + 7)^{3} \\
216x - 16 & = (14742x^{2} - 2157x + 82)^{3} + (-14742x^{2} + 2211x - 86)^{3} + (4914x^{2} - 971x + 44)^{3} + (-4914x^{2} + 485x - 8)^{3} \\
216x + 92 & = (3x - 164)^{3} + (-3x + 160)^{3} + (x - 35)^{3} + (-x + 71)^{3}
\end{align}</math>

Together with their complementary identities, these prove the 18x±2 case with the exception of integers congruent to 108x±38. Demjanenko also proves the 108x±38 case in his paper using more advanced methods, thus completely proving the 18x±2 case.<ref name="Demjanenko 1966"/>

By considering this result along with the earlier results from Sierpiński and Schinzel, only the 18x±4 and 18x±5 cases remain unproven, which can be expressed more simply as 9x±4.

Demjanenko also provides a table of decompositions of many specific numbers in the 9x±4 case, which he states expands on the work of Sierpiński and Schinzel to prove all integers with absolute value less than or equal to 1000 may be decomposed into the sum of 4 cubes.<ref name="Demjanenko 1966"/>

== See also ==
* [Sums of three cubes](/source/Sums_of_three_cubes)

== Notes and references ==
{{reflist}}

Category:Diophantine equations
Category:Unsolved problems in number theory

---
Adapted from the Wikipedia article [Sum of four cubes problem](https://en.wikipedia.org/wiki/Sum_of_four_cubes_problem) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Sum_of_four_cubes_problem?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.