Reducing the Erdős-Moser equation \(1^n + 2^n + \dots + k^n = (k + 1)^n\) modulo \(k\) and \(k^2\). (English) Zbl 1233.11038

An open conjecture of Erdős and Moser (from around 1950) is that the only solution of the Diophantine equation \(1^n+2^n+\dots+k^n=(k+1)^n\) is the trivial solution \(1+2 = 3\). Y. Gallot, P. Moree and W. Zudilin [Math. Comput. 80, No. 274, 1221–1237 (2011; Zbl 1231.11038)] showed that if there is a further solution then both \(k\) and \(n\) must exceed \(10^{10^9}\). By reducing the equation modulo \(k^2\) the authors find some new conditions that solutions \((k,n)\) have to satisfy. The proofs use divisibility properties of power sums as well as Lerch’s relation between Fermat and Wilson quotients.


11D61 Exponential Diophantine equations
11D79 Congruences in many variables
11A41 Primes


Zbl 1231.11038
Full Text: DOI arXiv