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Week 11 · ErdősOpen
Primary source

Erdős–Moser Equation — Is 1+2=31 + 2 = 3 the Only Solution?

Research level· Posed 1953

Problem

Find all natural-number solutions of 1k+2k+⋯+(m−1)k=mk1^k + 2^k + \cdots + (m-1)^k = m^k. The only known solution is m=3,k=1m=3, k=1 (1+2=31+2=3). Erdős conjectured this is the only solution.

Why it matters

Posed by Erdős (~1953). Motivation: a finite sum equals a single power — striking equality. For k=1k=1 there should be many solutions, but in fact only m=3m=3. For k≥2k \ge 2, none are expected — but candidate counterexamples must be astronomical.

Progress so far

If another solution (m,k)(m, k) exists, then m>10109m > 10^{10^9} (Gallot, Moree, Zudilin 2010). This astronomical lower bound comes from divisibility properties of Bernoulli numbers and deep congruences. Strong heuristic for nonexistence, but not a proof.

Further reading

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