Counterexample to Euler's conjecture on sums of like powers¶
Lander, L. J., & Parkin, T. R. (1966). Counterexample to Euler's conjecture on sums of like powers. Bulletin of the American Mathematical Society, 72(6).
Cited by¶
2 citations across 2 artifacts.
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Primes¶
- Negative Case Analysis
- Take the conjecture (Euler's, 1769) that no \(n\)th power is the sum of fewer than \(n\) smaller \(n\)th powers — for \(n=4\), that no fourth power equals the sum of three fourth powers.
This sourceReports the first computer-found counterexample (27^5 + 84^5 + 110^5 + 133^5 = 144^5) refuting Euler's sum-of-like-powers conjecture — the deliberate counterexample hunt against a universal claim.
- Take the conjecture (Euler's, 1769) that no \(n\)th power is the sum of fewer than \(n\) smaller \(n\)th powers — for \(n=4\), that no fourth power equals the sum of three fourth powers.
Mechanisms¶
- Universal Counterexample Test
- Euler's own sum-of-powers conjecture met exactly this fate: it stood for two centuries on confirming cases until a direct search turned up 27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵, a single counterexample that ended it.
This sourceReports the direct computer search that found 27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵, refuting Euler’s sum-of-powers conjecture.
- Euler's own sum-of-powers conjecture met exactly this fate: it stood for two centuries on confirming cases until a direct search turned up 27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵, a single counterexample that ended it.
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