Euler's Sum of Powers Conjecture

Euler's Sum Of Powers Conjecture

Euler's conjecture is a disproved 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 greater than or equal to k.

In symbols, if \sum_{i=1}^{n} a_i^k = b^k where and are positive integers, then .

The conjecture represents an attempt to generalization of Fermat's last theorem, which could be seen as the special case of n = 2: if, then .

Although the conjecture holds for the case of k = 3 (which follows from Fermat's last theorem for the third powers), it was disproved for k = 4 and k = 5. It still remains unknown if the conjecture fails or holds for any value k ≥ 6.

Read more about Euler's Sum Of Powers Conjecture:  Generalizations

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