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 kth powers of positive integers is itself a kth 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 $n\geq k$.

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 $a_{i}\neq b_{j}$ are positive integers for all $1\leq i\leq n$ and $1\leq j\leq m$, then $m+n\geq k.$

See also

• Euler's equation of degree four

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!
• Mathematicians Find New Solutions To An Ancient Puzzle