Concept# Euler's sum of powers conjecture

Summary

Euler's conjecture is a disproved conjecture in mathematics related to Fermat's Last Theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers n and k greater than 1, if the sum of n many kth powers of positive integers is itself a kth power, then n is greater than or equal to k:
:a + a + ... + a = bk ⇒ n ≥ k
The conjecture represents an attempt to generalize Fermat's Last Theorem, which is the special case n = 2: if a + a = bk, then 2 ≥ k.
Although the conjecture holds for the case k = 3 (which follows from Fermat's Last Theorem for the third powers), it was disproved for k = 4 and k = 5. It is unknown whether the conjecture fails or holds for any value k ≥ 6.
Background
Euler was aware of the equality 594 + 1584 = 1334 + 1344 involving sums of four fourth powers; this, however, is not a counterex

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