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=                  Euler's sum of powers conjecture                  =
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                             Introduction                             
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In number theory, Euler's conjecture is a disproved conjecture related
to Fermat's Last Theorem.  It was proposed by Leonhard Euler in 1769.
It states that for all integers  and  greater than 1, if the sum of
many th powers of positive integers is itself a th power, then  is
greater than or equal to :



The conjecture represents an attempt to generalize Fermat's Last
Theorem, which is the special case : if  then .

Although the conjecture holds for the case  (which follows from
Fermat's Last Theorem for the third powers), it was disproved for  and
.. It is unknown whether the conjecture fails or holds for any value .


                              Background                              
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Euler was aware of the equality  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  or the taxicab number 1729.
The general solution of the equation
is



where ,  and  are any rational numbers.


                           Counterexamples                            
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Euler's conjecture was disproven by L. J. Lander and T. R. Parkin in
1966 when, through a direct computer search on a CDC 6600, they found
a counterexample for . This was published in a paper comprising just
two sentences. A total of three primitive (that is, in which the
summands do not all have a common factor) counterexamples are known:

(Lander & Parkin, 1966); (Scher & Seidl, 1996); (Frye, 2004).

In 1988, Noam Elkies published a method to construct an infinite
sequence of counterexamples for the  case. His smallest counterexample
was


A particular case of Elkies' solutions can be reduced to the identity

where

This is an elliptic curve with a rational point at . From this initial
rational point, one can compute an infinite collection of others.
Substituting  into the identity and removing common factors gives the
numerical example cited above.

In 1988, Roger Frye found the smallest possible counterexample

for  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.


                           Generalizations                            
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In 1967, L. J. Lander, T. R. Parkin, and John Selfridge conjectured
that if
:,
where  are positive integers for all  and , then .  In the special
case , the conjecture states that if
:
(under the conditions given above) then .

The special case may be described as the problem of giving a partition
of a perfect power into few like powers.  For  and  or , there are
many known solutions.  Some of these are listed below.

See  for more data.

======
(Plato's number 216)
This is the case ,  of Srinivasa Ramanujan's formula


A cube as the sum of three cubes can also be parameterized in one of
two ways:

The number 2,100,0003 can be expressed as the sum of three positive
cubes in nine different ways.

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(R. Frye, 1988); (R. Norrie, smallest, 1911).

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(Lander & Parkin, 1966); (Lander, Parkin, Selfridge, smallest,
1967); (Lander, Parkin, Selfridge, second smallest, 1967); (Sastry,
1934, third smallest).

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It has been known since 2002 that there are no solutions for  whose
final term is ≤ 730000.

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(M. Dodrill, 1999).

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(S. Chase, 2000).


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


                            External links                            
======================================================================
* Tito Piezas III, [http://sites.google.com/site/tpiezas/Home/  A
Collection of Algebraic Identities]
* 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)]
* R. Gerbicz, J.-C. Meyrignac, U. Beckert,
[https://arxiv.org/abs/1108.0462 All solutions of the Diophantine
equation 'a'6 + 'b'6 = 'c'6 + 'd'6 + 'e'6 + 'f'6 + 'g'6 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]
*
*
*
*
[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!


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