- Euler's sum of powers conjecture
Euler's conjecture is a disproved
conjecturein mathematicsrelated to Fermat's last theoremwhich was proposed by Leonhard Eulerin 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, ifwhere and are positive integers, then .
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.
Noam Elkiesfound a method to construct counterexamples for the "k" = 4 case. His smallest counterexample was the following:
::26824404 + 153656394 + 187967604 = 206156734.
Roger Fryesubsequently 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 Selfridgeconjectured that for every ,if , where are positive integers for all and , then
Euler's equation of degree four
* [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]
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