# Euler Catches and get Caught

The $n$th Fermat number is defines as:

$F_n = 2^{2^n} + 1$

The 17th-Century French lawyer and mathematician Pierre de Fermat incorrectly conjectured that all these numbers were primes, although he had no proof. The first 5 Fermat numbers: $3, 5, 17, 257, 65537$ corresponding to $n = 0,1,2,3,4$ are all primes. These are called Fermat primes. Leonhard Euler proved in 1732 that 641 is divisor of $F_5$. Moreover, no other Fermat number is known to be prime for $n > 4$. Now it is conjectured that those are all prime Fermat numbers. It is also unknown whether there are infinitely many composite Fermat numbers or not.

Pierre de Fermat had proved that no set of positive integers satisfy the equation $a^4+b^4=c^4$. Euler conjectured that the equation $a^4+b^4+c^4=d^4$ would also have no integer solution. Euler went further. He proposed that for every integer greater than 2, the sum of $n-1$ $n$th powers of positive integers can not itself be an nth power. For example sum of three 4th powers can not be another 4th power.

The conjecture was disproven by L. J. Lander and T. R. Parkin in 1966 when they found the following counterexample using direct computer search for $n = 5:$

$27^5+84^5+110^5+133^5=144^5$

Yet another counter example was found by Jim Frye in 2004.

$85282^5+28969^5+3183^5+55^5=85359^5$

In 1986, Noam Elkies found a method to construct an infinite series counterexamples for the $n=4$ case. His smallest counterexample was the follwing.

$2682440^4+15365639^4+18796760^4=20615673^4$

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

$95800^4+217519^4+414560^4=422481^4$