Euler Quartic Conjecture/Historical Note
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Historical Note on Euler Quartic Conjecture
Leonhard Paul Euler put forward this conjecture in $1772$.
Nobody made any progress on proving it one way or another until Noam David Elkies discovered the counterexample $20 \, 615 \, 673$ in $1987$, and proved that there exists an infinite number of such solutions.
Soon after that, Roger Frye found the smaller counterexample $422 \, 481$, and demonstrated that there were none smaller.
Both counterexamples were published in the cited article in Mathematics of Computation by Elkies in $1988$.
This discovery was subsequently reported in the New York Times.
Allan MacLeod found the solution $638 \, 523 \, 249$ in $1997$.
Sources
- Oct. 1988: Noam D. Elkies: On $A^4 + B^4 + C^4 = D^4$ (Math. Comp. Vol. 51, no. 184: pp. 825 – 835) www.jstor.org/stable/2008781
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $20,615,673$
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Euler's conjecture
- Weisstein, Eric W. "Diophantine Equation--4th Powers." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DiophantineEquation4thPowers.html
- Weisstein, Eric W. "Euler Quartic Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulerQuarticConjecture.html