Euler's sum of powers 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, 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 $a_i\ne b_j$ are positive integers for all $1\leq i\leq n$ and $1\leq j\leq m$ , then $m + n \geq k.$