# Euler's Sum of Powers Conjecture

**Type:** Disproven Conjecture  
**Domain:** named-results  
**Codex URL:** /mathematics/named-results/eulers-sum-of-powers-conjecture/  
**Entry status:** Live — v1.0 (2026-07-09)

## Summary
Conjectured 1769: at least n nth powers needed to sum to an nth power. Disproven for k=5 by Lander-Parkin 1966 (27⁵+84⁵+110⁵+133⁵=144⁵) and k=4 by Elkies 1988. One of the shortest published counterexample papers in mathematics.

## Sources
- [Tier 1] Lander, L.J. and Parkin, T.R. (1966). Bull. AMS, 72(6), 1079.
- [Tier 1] Elkies, N.D. (1988). On A⁴+B⁴+C⁴=D⁴. Mathematics of Computation, 51(184), 825-835.
- [Tier 2] Euler, L. (1769). Letter to Goldbach, 17 April 1769.
- [Tier 3] Guy, R.K. (2004). Unsolved Problems in Number Theory. 3rd ed. Section D1.

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*Mathematics Codex entry v1.0 — added 2026-07-09 — thecodex.expert/mathematics/*