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Erdős-Moser Equation

The Diophantine equation

sum_(j=1)^(m-1)j^n=m^n.

Erdős conjectured that there is no solution to this equation other than the trivial solution 1^1+2^1=3^1, although this remains unproved (Guy 1994, pp. 153-154). Moser (1953) proved that there is no solution for m<10^(10^6), and Butske et al. (1999) extended this to m<10^(9.3×10^6), or more specifically, m<1.485×10^(9321155).

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References

Butske, W.; Jaje, L. M.; and Mayernik, D. R. "The Equation sum_(p|N)1/p+1/N=1, Pseudoperfect Numbers, and Partially Weighted Graphs." Math. Comput. 69, 407-420, 1999.Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994.Moree, P. "Diophantine Equations of Erdős-Moser Type." Bull. Austral. Math. Soc. 53, 281-292, 1996.Moser, L. "On the Diophantine Equation 1^n+2^n+3^n+...+(m-1)^n=m^n." Scripta Math. 19, 84-88, 1953.

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Erdős-Moser Equation

Cite this as:

Weisstein, Eric W. "Erdős-Moser Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Erdos-MoserEquation.html

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