View source for Lander, Parkin, and Selfridge conjecture

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

In [[number theory]], the '''Lander, Parkin, and Selfridge conjecture''' concerns the [[integer]] solutions of equations which contain sums of like [[Exponentiation|powers]].  The equations are generalisations of those considered in [[Fermat's Last Theorem]]. The conjecture is that if the sum of some {{mvar|k}}-th powers equals the sum of some other {{mvar|k}}-th powers, then the total number of terms in both sums combined must be at least {{mvar|k}}.

== Background ==
[[Diophantine equation]]s, such as the integer version of the equation {{math|1=''a''<sup>2</sup> + ''b''<sup>2</sup> = ''c''<sup>2</sup>}} that appears in the [[Pythagorean theorem]], have been studied for their [[integer]] [[Solution (equation)|solution]] properties for centuries. [[Fermat's Last Theorem]] states that for [[Exponentiation|power]]s greater than 2, the equation {{math|1=''a''<sup>''k''</sup> + ''b''<sup>''k''</sup> = ''c''<sup>''k''</sup>}} has no solutions in non-zero [[integer]]s  {{math|''a'', ''b'', ''c''}}. Extending the number of [[Summand|term]]s on either or both sides, and allowing for higher powers than 2, led to [[Leonhard Euler]] to propose in 1769 that for all integers {{mvar|n}} and {{mvar|k}} greater than 1, if the sum of {{math|''n'' ''k''}}th powers of positive integers is itself a {{mvar|k}}th power, then {{mvar|n}} is greater than or equal to {{mvar|k}}.

In symbols, if
<math>
\sum_{i=1}^{n} a_i^k = b^k
</math>
where {{math|''n'' > 1}} and <math>a_1, a_2, \dots, a_n, b</math> are positive integers, then his conjecture was that {{math|''n'' ≥ ''k''}}.

In [[1966 in science|1966]], a counterexample to [[Euler's sum of powers conjecture]] was found by [[Leon J. Lander]] and [[Thomas R. Parkin]] for {{math|1=''k'' = 5}}:<ref>{{cite journal |author1=L. J. Lander |author2=T. R. Parkin |title=Counterexample to Euler's conjecture on sums of like powers |journal=Bull. Amer. Math. Soc. |volume=72 |year=1966 |page=1079 |doi=10.1090/S0002-9904-1966-11654-3|doi-access=free }}</ref>
:{{math|1=27<sup>5</sup> + 84<sup>5</sup> + 110<sup>5</sup> + 133<sup>5</sup> = [[144 (number)|144]]<sup>5</sup>}}.

In subsequent years, further [[Euler's sum of powers conjecture#Counterexamples|counterexamples]] were found, including for {{math|1=''k'' = 4}}. The latter disproved the more specific [[Euler quartic conjecture]], namely that {{math|1=''a''<sup>4</sup> + ''b''<sup>4</sup> + ''c''<sup>4</sup> = ''d''<sup>4</sup>}} has no positive integer solutions.  In fact, the smallest solution, found in 1988, is

:{{math|1=414560<sup>4</sup> + 217519<sup>4</sup> + 95800<sup>4</sup> = 422481<sup>4</sup>}}.

== Conjecture ==

In 1967, L. J. Lander, T. R. Parkin, and [[John Selfridge]] conjectured<ref>{{cite journal |author1=L. J. Lander |author2=T. R. Parkin |author3=J. L. Selfridge |title=A Survey of Equal Sums of Like Powers |journal=[[Mathematics of Computation]] |volume=21 |issue=99 |year=1967 |pages=446–459 |doi=10.1090/S0025-5718-1967-0222008-0 |jstor=2003249|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''}}. The equal sum of like powers formula is often abbreviated as&nbsp;{{math|(''k'', ''m'', ''n'')}}.

Small examples with <math>m=n=\tfrac{k}{2}</math> (related to [[generalized taxicab number]]) include 
:{{math|1=59{{sup|4}} + 158{{sup|4}} = 133{{sup|4}} + 134{{sup|4}}}} (known to Euler) 
and 
:{{math|1=3{{sup|6}} + 19{{sup|6}} + 22{{sup|6}} = 10{{sup|6}} + 15{{sup|6}} + 23{{sup|6}}}} (found by K. Subba Rao in 1934).

The conjecture implies in the special case of {{math|1=''m'' = 1}} that if 
<math display="block">\sum_{i=1}^n a_i^k = b^k</math>
(under the conditions given above) then {{math|''n'' ≥ ''k'' − 1}}.

[[File:Plato_number.svg|thumb|An interpretation of [[Plato's number]] is a solution for {{math|1=''k'' = 3}}]]
For this special case of {{math|1=''m'' = 1}}, some of the known solutions satisfying the proposed constraint with {{math|''n'' ≤ ''k''}}, where terms are [[positive integer]]s, hence giving a [[integer partition|partition]] of a power into like powers, are:<ref name=meyrignac>Quoted in {{cite web
| last = Meyrignac
| first = Jean-Charles
| url = http://euler.free.fr/records.htm
| title = Computing Minimal Equal Sums Of Like Powers: Best Known Solutions
| date = 14 February 2001
| accessdate = 17 July 2017
}}</ref>

;{{math|1=''k'' = 3}}
::{{math|1=3<sup>3</sup> + 4<sup>3</sup> + 5<sup>3</sup> = 6<sup>3</sup>}}

;{{math|1=''k'' = 4}}
::{{math|1=95800<sup>4</sup> + 217519<sup>4</sup> + 414560<sup>4</sup> = 422481<sup>4</sup>}}  (Roger Frye, 1988)

::{{math|1=30<sup>4</sup> + 120<sup>4</sup> + 272<sup>4</sup> + 315<sup>4</sup> = 353<sup>4</sup>}} (R. Norrie, 1911)
Fermat's Last Theorem implies that for {{math|1=''k'' = 4}} the conjecture is true.

;{{math|1=''k'' = 5}}
::{{math|1=27<sup>5</sup> + 84<sup>5</sup> + 110<sup>5</sup> + 133<sup>5</sup> = 144<sup>5</sup>}} (Lander, Parkin, 1966)

::{{math|1=7<sup>5</sup> + 43<sup>5</sup> + 57<sup>5</sup> + 80<sup>5</sup> + 100<sup>5</sup> = 107<sup>5</sup>}} (Sastry, 1934, third smallest)

;{{math|1=''k'' = 6}}
::(None known. As of 2002, there are no solutions whose final term is {{math|≤ 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|1=''k'' = 7}}
::{{math|1=127<sup>7</sup> + 258<sup>7</sup> + 266<sup>7</sup> + 413<sup>7</sup> + 430<sup>7</sup> + 439<sup>7</sup> + 525<sup>7</sup> = 568<sup>7</sup>}} (M. Dodrill, 1999)

;{{math|1=''k'' = 8}}
::{{math|1=90<sup>8</sup> + 223<sup>8</sup> + 478<sup>8</sup> + 524<sup>8</sup> + 748<sup>8</sup> + 1088<sup>8</sup> + 1190<sup>8</sup> + 1324<sup>8</sup> = 1409<sup>8</sup>}}  (Scott Chase, 2000)

;{{math|''k'' ≥ 9}}
::(None known.)

== Current status==
It is not known if the conjecture is true, or if nontrivial solutions exist that would be counterexamples, such as {{math|1=''a''<sup>''k''</sup> + ''b''<sup>''k''</sup> = ''c''<sup>''k''</sup> + ''d''<sup>''k''</sup>}} for {{math|1=''k'' ≥ 5}}.
<ref>{{cite journal |author1=A. Bremner |author2=R. K. Guy |title=A Dozen Difficult Diophantine Dilemmas |journal=[[American Mathematical Monthly]] |volume=95 |issue=1 |year=1998 |pages=31–36 |doi=10.2307/2323442 |jstor=2323442 |doi-access= }}</ref>
<ref>{{cite journal |author1=T.D. Browning |title=Equal sums of two kth powers. |journal=[[Journal of Number Theory]] |volume=96 |issue=2 |year=2002 |pages=293-318}}</ref>

==See also==
*[[Beal's conjecture]]
*[[Fermat–Catalan conjecture]]
*[[Jacobi–Madden equation]]
*[[List of unsolved problems in mathematics]]
*[[Minimum Sudoku problem|Experimental mathematics]] (counterexamples to Euler's sum of powers conjecture, especially smallest solution for ''k'' = 4)
*[[Prouhet–Tarry–Escott problem]]
*[[Pythagorean quadruple]]
*[[Sums of powers]], a list of related conjectures and theorems

== References ==
{{reflist}}
* {{cite book | first=Richard K. | last=Guy | authorlink=Richard K. Guy |title=Unsolved Problems in Number Theory | edition=3rd | publisher=[[Springer-Verlag]] | year=2004 | isbn=0-387-20860-7 | zbl=1058.11001 | series=Problem Books in Mathematics | location=New York, NY | at=D1 }}

== External links ==
* [http://euler.free.fr/ EulerNet: Computing Minimal Equal Sums Of Like Powers]
* Jaroslaw Wroblewski [http://www.math.uni.wroc.pl/~jwr/eslp/  Equal Sums of Like Powers]
* Tito Piezas III: [http://sites.google.com/site/tpiezas/Home/  A Collection of Algebraic Identities]
* {{MathWorld |title=Diophantine Equation--5th Powers |urlname=DiophantineEquation5thPowers}}
* {{MathWorld |title=Diophantine Equation--6th Powers |urlname=DiophantineEquation6thPowers}}
* {{MathWorld |title=Diophantine Equation--7th Powers |urlname=DiophantineEquation7thPowers}}
* {{MathWorld |title=Diophantine Equation--8th Powers |urlname=DiophantineEquation8thPowers}}
* {{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/20080603072136/http://library.thinkquest.org/28049/Euler's%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!
* [https://www.sciencedaily.com/releases/2008/03/080314145039.htm Mathematicians Find New Solutions To An Ancient Puzzle]
* Ed Pegg Jr. [https://web.archive.org/web/20120503192523/http://www.maa.org/editorial/mathgames/mathgames_11_13_06.html Power Sums], Math Games

[[Category:Diophantine equations]]
[[Category:Conjectures]]
[[Category:Unsolved problems in number theory]]