Euler's sum of powers conjecture

Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem which was proposed by Leonhard Euler in 1769. It states that for all integers "n" and "k" greater than 1, if the sum of "n" "k"th powers of positive integers is itself a "k"th power, then "n" is not smaller than "k".

In symbols, if$sum\_\{i=1\}^\{n\}\; a\_i^k\; =\; b^k$where $n>1$ and $a\_1,\; a\_2,\; dots,\; a\_n,\; b$ are positive integers, then $ngeq\; k$.

The conjecture was disproven by L. J. Lander and T. R. Parkin in 1966 when they found the following counterexample for "k" = 5:

::27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵.

In 1986, Noam Elkies found a method to construct counterexamples for the "k" = 4 case. His smallest counterexample was the following:

::2682440⁴ + 15365639⁴ + 18796760⁴ = 20615673⁴.

In 1988, Roger Frye subsequently found the smallest possible "k" = 4 counterexample by a direct computer search using techniques suggested by Elkies:

::95800⁴ + 217519⁴ + 414560⁴ = 422481⁴.

In 1966, L. J. Lander, T. R. Parkin, and John Selfridge conjectured that for every $k>3$,if $sum\_\{i=1\}^\{n\}\; a\_i^k\; =\; sum\_\{j=1\}^\{m\}\; b\_j^k$, where $a\_i\; e\; b\_j$ are positive integers for all $1leq\; ileq\; n$ and $1leq\; jleq\; m$, then $m\; +\; n\; geq\; k.$

ee also

*Euler's equation of degree four

External links

* [http://euler.free.fr/ EulerNet: Computing Minimal Equal Sums Of Like Powers]
* [http://mathworld.wolfram.com/EulerQuarticConjecture.html Euler Quartic Conjecture] at MathWorld
* [http://mathworld.wolfram.com/DiophantineEquation4thPowers.html Diophantine Equation — 4th Powers] at MathWorld
* [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!
* [http://www.sciencedaily.com/releases/2008/03/080314145039.htm Mathematicians Find New Solutions To An Ancient Puzzle]