Number Theory


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

Authors: Arkoprobho Chakraborty

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.

Comments: 13 pages.

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Submission history

[v1] 12 Dec 2009

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