Editing Euler's sum of powers conjecture

{{short description|Disproved conjecture in number theory}} 

In [[number theory]], '''Euler's conjecture''' is a [[List of disproved mathematical ideas|disproved]] [[conjecture]] related to [[Fermat's Last Theorem]].  It was proposed by [[Leonhard Euler]] in 1769. It states that for all [[integers]] {{mvar|n}} and {{mvar|k}} greater than 1, if the sum of {{mvar|n}} many {{mvar|k}}th powers of positive integers is itself a {{mvar|k}}th power, then {{mvar|n}} is greater than or equal to {{mvar|k}}:

<math display=block>a_1^k + a_2^k + \dots + a_n^k = b^k \implies n \ge k</math>

The conjecture represents an attempt to generalize Fermat's Last Theorem, which is the special case {{math|''n'' {{=}} 2}}: if <math>a_1^k + a_2^k = b^k,</math> then {{math|2 ≥ ''k''}}.

Although the conjecture holds for the case {{math|''k'' {{=}} 3}} (which follows from Fermat's Last Theorem for the third powers), it was disproved for {{math|''k'' {{=}} 4}} and {{math|''k'' {{=}} 5}}. It is unknown whether the conjecture fails or holds for any value {{math|''k'' ≥ 6}}.

== Background ==
Euler was aware of the equality {{nowrap|59{{sup|4}} + 158{{sup|4}} {{=}} 133{{sup|4}} + 134{{sup|4}}}} involving sums of four fourth powers; this, however, is not a [[counterexample]] because no term is isolated on one side of the equation. He also provided a complete solution to the four cubes problem as in [[Plato's number]] {{nowrap|3{{sup|3}} + 4{{sup|3}} + 5{{sup|3}} {{=}} 6{{sup|3}}}} or the [[taxicab number]] 1729.<ref>{{cite book |editor-last=Dunham |editor-first=William |year=2007 |title=The Genius of Euler: Reflections on His Life and Work |publisher=The MAA |isbn=978-0-88385-558-4 |page=220 |url=https://books.google.com/books?id=M4-zUnrSxNoC&pg=PA220}}</ref><ref>{{cite web |last=Titus, III |first=Piezas |year=2005 |title=Euler's Extended Conjecture |url=http://www.oocities.org/titus_piezas/Equalsums.htm }}</ref> The general solution of the equation <math>x_1^3+x_2^3=x_3^3+x_4^3</math>
is

<math display=block>\begin{align}
  x_1 &=\lambda( 1-(a-3b)(a^2+3b^2)) \\[2pt]
  x_2 &=\lambda( (a+3b)(a^2+3b^2)-1 )\\[2pt]
  x_3 &=\lambda( (a+3b)-(a^2+3b^2)^2 )\\[2pt]
  x_4 &= \lambda( (a^2+3b^2)^2-(a-3b))
\end{align}</math>

where {{mvar|a}}, {{mvar|b}} and <math>{\lambda}</math> are any rational numbers.

== Counterexamples ==
Euler's conjecture was disproven by [[Leon J. Lander|L. J. Lander]] and [[Thomas R. Parkin|T. R. Parkin]] in 1966 when, through a direct computer search on a [[CDC 6600]], they found a counterexample for {{math|''k'' {{=}} 5}}.<ref name="LanderParkin">{{cite journal |last1=Lander |first1=L. J. |last2=Parkin |first2=T. R. |year=1966 |title=Counterexample to Euler's conjecture on sums of like powers |journal=Bull. Amer. Math. Soc. |doi=10.1090/S0002-9904-1966-11654-3 |volume=72 |issue=6 |page=1079|doi-access=free }}</ref> This was published in a paper comprising just two sentences.<ref name="LanderParkin" /> A total of three primitive (that is, in which the summands do not all have a common factor) counterexamples are known:
<math display=block>\begin{align}
  144^5 &= 27^5 + 84^5 + 110^5 + 133^5 \\
  14132^5 &= (-220)^5 + 5027^5 + 6237^5 + 14068^5 \\
  85359^5 &= 55^5 + 3183^5 + 28969^5 + 85282^5
\end{align}</math>
(Lander & Parkin, 1966); (Scher & Seidl, 1996); (Frye, 2004).

In 1988, [[Noam Elkies]] published a method to construct an infinite sequence of counterexamples for the {{math|''k'' {{=}} 4}} case.<ref name="Elkies1988">{{cite journal |last=Elkies |first=Noam |authorlink=Noam Elkies |year=1988 |title=On ''A''<sup>4</sup> + ''B''<sup>4</sup> + ''C''<sup>4</sup> = ''D''<sup>4</sup> |journal=[[Mathematics of Computation]] |doi=10.1090/S0025-5718-1988-0930224-9 |mr=0930224 |jstor=2008781 |volume=51 |issue=184 |pages=825–835 |url=https://www.ams.org/journals/mcom/1988-51-184/S0025-5718-1988-0930224-9/S0025-5718-1988-0930224-9.pdf |doi-access=free }}</ref> His smallest counterexample was
<math display=block>20615673^4 = 2682440^4 + 15365639^4 + 18796760^4.</math>

A particular case of Elkies' solutions can be reduced to the identity<ref>{{cite web |title=Elkies' ''a''<sup>4</sup>+''b''<sup>4</sup>+''c''<sup>4</sup> = ''d''<sup>4</sup> |url=https://groups.google.com/group/sci.math/browse_thread/thread/15beef75eaddcb1b?hl=en#}}</ref><ref>{{cite book|year = 2010|chapter = Sums of Three Fourth Powers (Part 1)|title = A Collection of Algebraic Identities|first = Tito|last = Piezas III|chapter-url = http://sites.google.com/site/tpiezas/014|access-date = April 11, 2022}}</ref>
<math display=block>(85v^2 + 484v - 313)^4 + (68v^2 - 586v + 10)^4 + (2u)^4 = (357v^2 - 204v + 363)^4,</math>
where
<math display=block>u^2 = 22030 + 28849v - 56158v^2 + 36941v^3 - 31790v^4.</math>
This is an [[elliptic curve]] with a [[rational point]] at {{math|''v''<sub>1</sub> {{=}} −{{sfrac|31|467}}}}. From this initial rational point, one can compute an infinite collection of others. Substituting {{math|''v''<sub>1</sub>}} into the identity and removing common factors gives the numerical example cited above.

In 1988, [[Roger Frye]] found the smallest possible counterexample 
<math display=block>95800^4 + 217519^4 + 414560^4 = 422481^4</math>
for {{math|''k'' {{=}} 4}} by a direct computer search using techniques suggested by Elkies. This solution is the only one with values of the variables below 1,000,000.<ref>{{citation
 | last = Frye | first = Roger E.
 | year = 1988
 | title = Proceedings of Supercomputing 88, Vol.II: Science and Applications
 | contribution = Finding 95800<sup>4</sup> + 217519<sup>4</sup> + 414560<sup>4</sup> = 422481<sup>4</sup> on the Connection Machine
 | doi = 10.1109/SUPERC.1988.74138
 | pages = 106–116| s2cid = 58501120
 }}</ref>

== Generalizations ==
[[File:Plato_number.svg|thumb|250px|One interpretation of Plato's number, {{nowrap|3<sup>3</sup> + 4<sup>3</sup> + 5<sup>3</sup> {{=}} 6<sup>3</sup>}}]]
{{main article|Lander, Parkin, and Selfridge conjecture}}
In 1967, L. J. Lander, T. R. Parkin, and [[John Selfridge]] conjectured<ref name="autogenerated446">{{cite journal |last1=Lander |first1=L. J. |last2=Parkin |first2=T. R. |last3=Selfridge |first3=J. L. |year=1967 |title=A Survey of Equal Sums of Like Powers |journal=[[Mathematics of Computation]] |doi=10.1090/S0025-5718-1967-0222008-0 |jstor=2003249 |volume=21 |issue=99 |pages=446–459 |doi-access=free }}</ref> that if 
:<math>\sum_{i=1}^{n} a_i^k = \sum_{j=1}^{m} b_j^k</math>,
where {{math|''a<sub>i</sub>'' ≠ ''b<sub>j</sub>''}} are positive integers for all {{math|1 ≤ ''i'' ≤ ''n''}} and {{math|1 ≤ ''j'' ≤ ''m''}}, then {{math|''m'' + ''n'' ≥ ''k''}}.  In the special case {{math|''m'' {{=}} 1}}, the conjecture states that if
:<math>\sum_{i=1}^{n} a_i^k = b^k</math>
(under the conditions given above) then {{math|''n'' ≥ ''k'' − 1}}.

The special case may be described as the problem of giving a [[integer partition|partition]] of a perfect power into few like powers.  For {{math|''k'' {{=}} 4, 5, 7, 8}} and {{math|''n'' {{=}} ''k''}} or {{math|''k'' − 1}}, there are many known solutions.  Some of these are listed below. 

See {{OEIS2C|A347773}} for more data.

==={{math|''k'' {{=}} 3}}===
<math display="block">3^3 + 4^3 + 5^3 = 6^3</math> ([[Plato's number]] 216)
This is the case {{math|1=''a'' = 1}}, {{math|1=''b'' = 0}} of [[Srinivasa Ramanujan]]'s formula<ref name=Weisstein>{{cite web| url = http://mathworld.wolfram.com/DiophantineEquation3rdPowers.html| title = MathWorld : Diophantine Equation--3rd Powers}}</ref> 
<math display=block>(3a^2+5ab-5b^2)^3 + (4a^2-4ab+6b^2)^3 + (5a^2-5ab-3b^2)^3 = (6a^2-4ab+4b^2)^3</math>

A cube as the sum of three cubes can also be parameterized in one of two ways:<ref name=Weisstein/>
<math display=block>\begin{align}
a^3(a^3+b^3)^3	&=	b^3(a^3+b^3)^3+a^3(a^3-2b^3)^3+b^3(2a^3-b^3)^3 \\[6pt]
a^3(a^3+2b^3)^3	&=	a^3(a^3-b^3)^3+b^3(a^3-b^3)^3+b^3(2a^3+b^3)^3.
\end{align}</math>
The number 2,100,000<sup>3</sup> can be expressed as the sum of three positive cubes in nine different ways.<ref name=Weisstein/>

==={{math|''k'' {{=}} 4}}===
<math display=block>\begin{align}
  422481^4 &= 95800^4 + 217519^4 + 414560^4 \\[4pt]
  353^4 &= 30^4 + 120^4 + 272^4 + 315^4
\end{align}</math>
(R. Frye, 1988);<ref name="Elkies1988"/> (R. Norrie, smallest, 1911).<ref name="autogenerated446"/>

==={{math|''k'' {{=}} 5}}===
<math display=block>\begin{align}
  144^5  &= 27^5 + 84^5 + 110^5 + 133^5 \\[2pt]
  72^5 &= 19^5 + 43^5 + 46^5 + 47^5 + 67^5 \\[2pt]
  94^5 &= 21^5 + 23^5 + 37^5 + 79^5 + 84^5 \\[2pt] 
  107^5 &= 7^5 + 43^5 + 57^5 + 80^5 + 100^5
\end{align}</math>

(Lander & Parkin, 1966);<ref name="mathologer">{{cite web |url=https://www.youtube.com/watch?v=AO-W5aEJ3Wg | archive-url=https://ghostarchive.org/varchive/youtube/20211211/AO-W5aEJ3Wg| archive-date=2021-12-11 | url-status=live|title=Euler's and Fermat's last theorems, the Simpsons and CDC6600 |author=Burkard Polster |website=[[YouTube]] |date=March 24, 2018 |author-link=Burkard Polster <!-- |work=[[Mathologer]] --> |type=video |access-date=2018-03-24}}{{cbignore}}</ref><ref>{{cite web| url = https://mathworld.wolfram.com/DiophantineEquation5thPowers.html| title = MathWorld: Diophantine Equation--5th Powers}}</ref><ref>{{cite web| url = https://pat7.com/jp/s515-10007-t| title = A Table of Fifth Powers equal to Sums of Five Fifth Powers}}</ref> (Lander, Parkin, Selfridge, smallest, 1967);<ref name="autogenerated446"/> (Lander, Parkin, Selfridge, second smallest, 1967);<ref name="autogenerated446"/> (Sastry, 1934, third smallest).<ref name="autogenerated446"/>

==={{math|''k'' {{=}} 6}}===
It has been known since 2002 that there are no solutions for {{math|1=''k'' = 6}} whose final term is ≤ 730000.<ref>Giovanni Resta and Jean-Charles Meyrignac (2002). [https://www.ams.org/journals/mcom/2003-72-242/S0025-5718-02-01445-X/S0025-5718-02-01445-X.pdf The Smallest Solutions to the Diophantine Equation <math>a^6+b^6+c^6+d^6+e^6=x^6+y^6</math>], Mathematics of Computation, v. 72, p. 1054 (See '''further work''' section).</ref>

==={{math|''k'' {{=}} 7}}===
<math display=block>568^7 = 127^7 + 258^7 + 266^7 + 413^7 + 430^7 + 439^7 + 525^7</math>

(M. Dodrill, 1999).<ref>{{cite web| url = https://mathworld.wolfram.com/DiophantineEquation7thPowers.html| title = MathWorld: Diophantine Equation--7th Powers}}</ref>

==={{math|''k'' {{=}} 8}}===
<math display=block>1409^8 = 90^8 + 223^8 + 478^8 + 524^8 + 748^8 + 1088^8 + 1190^8 + 1324^8 </math>

(S. Chase, 2000).<ref>{{cite web| url = https://mathworld.wolfram.com/DiophantineEquation8thPowers.html| title = MathWorld: Diophantine Equation--8th Powers}}</ref>

==See also==
* [[Jacobi–Madden equation]]
*[[Prouhet–Tarry–Escott problem]]
*[[Beal conjecture]]
*[[Pythagorean quadruple]]
*[[Generalized taxicab number]]
*[[Sums of powers]], a list of related conjectures and theorems

== References ==
{{reflist}}

== External links ==
* Tito Piezas III, [http://sites.google.com/site/tpiezas/Home/  A Collection of Algebraic Identities] {{Webarchive|url=https://web.archive.org/web/20111001021837/http://sites.google.com/site/tpiezas/Home |date=2011-10-01 }}
* Jaroslaw Wroblewski, [http://www.math.uni.wroc.pl/~jwr/eslp/  Equal Sums of Like Powers]
* Ed Pegg Jr., [https://web.archive.org/web/20080410224256/http://www.maa.org/editorial/mathgames/mathgames_11_13_06.html  Math Games, Power Sums]
* James Waldby, [http://pat7.com/jp/s515-10007-t A Table of Fifth Powers equal to a Fifth Power (2009)] 
* [[Robert Gerbicz|R. Gerbicz]], J.-C. Meyrignac, U. Beckert, [https://arxiv.org/abs/1108.0462 All solutions of the Diophantine equation ''a''<sup>6</sup> + ''b''<sup>6</sup> = ''c''<sup>6</sup> + ''d''<sup>6</sup> + ''e''<sup>6</sup> + ''f''<sup>6</sup> + ''g''<sup>6</sup> for ''a'',''b'',''c'',''d'',''e'',''f'',''g'' < 250000 found with a distributed Boinc project]
* [http://euler.free.fr/ EulerNet: Computing Minimal Equal Sums Of Like Powers]
* {{MathWorld |title=Euler's Sum of Powers Conjecture |urlname=EulersSumofPowersConjecture}}
* {{MathWorld |title=Euler Quartic Conjecture |urlname=EulerQuarticConjecture}}
* {{MathWorld |title=Diophantine Equation--4th Powers |urlname=DiophantineEquation4thPowers}}
* [https://web.archive.org/web/20071105172444/http://library.thinkquest.org/28049/Euler%27s%20conjecture.html Euler's Conjecture] at library.thinkquest.org
* [http://www.mathsisgoodforyou.com/conjecturestheorems/eulerconjecture.htm A simple explanation of Euler's Conjecture] at Maths Is Good For You!

{{Leonhard Euler}}

[[Category:Diophantine equations]]
[[Category:Disproved conjectures]]
[[Category:Leonhard Euler]]