Euler's conjecture
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Euler's conjecture is a 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 kth powers of positive integers is itself a kth power, then n is not smaller than k. In symbols, that is if 
where each ai's are some particular (positive) integer and b is another integer, 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.
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:
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- 958004 + 2175194 + 4145604 = 4224814.
In 1966 Lander, Parkin and John F. Selfridge conjectured that for every k > 3, if 
then 
[edit] External links
- EulerNet: Computing Minimal Equal Sums Of Like Powers
- Euler Quartic Conjecture at MathWorld
- Diophantine Equation — 4th Powers at MathWorld
- Euler's Conjecture at library.thinkquest.org
- A simple explanation of Euler's Conjecture at Maths Is Good For You!
