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

**Authors:** Arkoprobho Chakraborty

Erdos had conjectured that the equation of the title had no solutions
in natural numbers except the trivial 1^{1} + 2^{1} = 3^{1}. Moser (1953) had
shown that there are no solutions for M+1 < 10^{106}. Butske et al (1993)
had further shown that there are no solutions for M+1 < 9.3x10^{6}. 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.

**Comments:** 13 pages.

**Download:** **PDF**

### Submission history

[v1] 12 Dec 2009

**Unique-IP document downloads:** 131 times

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