Number Theory

Diophantine Equation 1^N + 2^N + ...+ (M 1)^N +M^N = (M + 1)^N

Authors: Arkoprobho Chakraborty
Comments: 13 pages.

Erdos had conjectured that the equation of the title had no solutions in natural numbers except the trivial 1¹ + 2¹ = 3¹. Moser (1953) had shown that there are no solutions for M+1 < 10¹⁰⁶. Butske et al (1993) had further shown that there are no solutions for M+1 < 9.3x10⁶. In this paper I show that a solution to this equation cannot exist for any value of M > 2 hence proving Erdos' conjecture. This is achieved using elementary number theoretic methods employing congruences and well-known identities.
Category: Number Theory