Let n,k≥ 2. Euler's sum of powers conjecture. If a_1^k+a_2^k+…+a_n^k=a_n+1^k, then n≥ k. The conjecture was refuted by a single counterexample, as reported in the source, so the claim is no longer open.

Euler's sum of powers conjecture

Euler proposed the assertion in 1778: representing one $k$ th power as a sum of $n$ $k$ th powers should require $n\geq k$ . Lander and Parkin disproved it in 1966 with an explicit exponent- $5$ example.

Known results

The assertion holds for $k=3$ .
Elkies found an exponent- $4$ counterexample in 1988; Frye found the smallest known one.
Lander--Parkin (1966) gave $27^5+84^5+110^5+133^5=144^5$ .
No counterexample for $k>5$ is reported in the cited literature.

February 24, 2026: new exponent- $5$ solution

Jeffrey Braun reported a fourth primitive solution for $k=5$ , namely $719115^5+1331622^5+(-1340632)^5+1956213^5=1956878^5$ . This adds another counterexample of the already-refuted type and does not address $k>5$ .

Current status (as of September 2026): The original conjecture is false by the Lander--Parkin counterexample, while whether counterexamples exist for $k>5$ remains open.

Euler's sum of powers conjecture

Primary source

Progress summary

Solutions 0