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. Moser 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.

Leo Moser | Paul Erdős