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 a_i's are some particular (positive) integer and b is another integer, then . In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Pierre de Fermats informal conjecture written in the margin of his book proved to be one of the most intriguing and enigmatic math problems ever devised. ... Leonhard Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ...

The conjecture was disproven by L. J. Lander and T. R. Parkin in 1966 when they found the following counterexample for k = 5: See also: Other events of 1966 List of years in science . ...

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: 144 is the whole number following 143 and preceding 145. ... Noam D. Elkies (born 1966 in New York City) is a mathematician. ...

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 Lander, Parkin and John F. Selfridge conjectured that for every k > 3, if then John Selfridge is a mathematician who has contributed to the field of analytic number theory. ...

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!