Disproved conjectures | Diophantine equations
Euler's sum of powers conjecture
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 k1 + a k2 + ... + a kn = bk ⇒ n ≥ k The conjecture represents an attempt to generalize Fermat's Last Theorem, which is the special case n = 2: if a k1 + a k2 = 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. (Wikipedia).
