Erdős–Moser equation

In number theory, the Erdős–Moser equation is

${\displaystyle 1^{k}+2^{k}+\cdots +(m-1)^{k}=m^{k},}$

where m and k are restricted to the positive integers—that is, it is considered as a Diophantine equation. The only known solution is 1¹ + 2¹ = 3¹, and Paul Erdős conjectured that no further solutions exist. Any further solutions must have m > 10¹⁰⁹.

Throughout this article, p refers exclusively to prime numbers.