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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 around 1953. For k=1k=1, the equation reduces to (m−1)m/2=m(m-1)m/2=m, giving the sole natural-number solution m=3m=3. For k≥2k \ge 2, no solution is expected, while congruence constraints force any counterexample to be astronomical.

Progress so far

If a solution with k>1k>1 exists, then m>2.7139×101,667,658,416m>2.7139\times10^{1,667,658,416} (Gallot–Moree–Zudilin, 2011). This is a rigorous lower bound derived from Bernoulli-number divisibility and many congruence conditions; it does not rule out solutions above the bound.

Further reading

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