The Shortest Paper That Shook Mathematics
In 1966, a paper of extraordinary brevity managed to overturn a mathematical belief that had stood for nearly two centuries. Published in a serious mathematical journal, it presented a single explicit counterexample to a conjecture made by Leonhard Euler.
Euler had conjectured that, for any power n > 2, at least n nth powers are required to sum to another nth power. In modern notation, he believed equations of the form
a₁ⁿ + a₂ⁿ + … + aₖⁿ = bⁿ
would require k ≥ n.
The Counterexample
Using a direct computer search on the CDC 6600, L. J. Lander and T. R. Parkin discovered the smallest known counterexample:
27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵
Here, four fifth powers sum to a single fifth power — directly contradicting Euler’s conjecture. That single equation was enough.
Why This Paper Is Famous
- It is one of the shortest papers ever published in a major mathematics journal.
- It refuted a conjecture proposed by one of history’s greatest mathematicians.
- It demonstrated the growing power of computational methods in pure mathematics.
- It reshaped how mathematicians view conjectures based on pattern and intuition.
Sometimes, mathematics does not advance through long proofs or complex theories, but through a single, perfectly chosen example.
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