Euler's Quartic Conjecture

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Famous False Conjecture

The Diophantine equation:

$A^4 = B^4 + C^4 + D^4$

has no solutions.


That is, there exist no integers $A, B, C, D \in \Z$ that satisfy the above.


Refutation

The following counterexamples exist:

\(\displaystyle 422 \, 481^4\) \(=\) \(\displaystyle 95 \, 800^4 + 217 \, 519^4 + 414 \, 560^4\)
\(\displaystyle 20 \, 615 \, 673^4\) \(=\) \(\displaystyle 2 \, 682 \, 440^4 + 15 \, 365 \, 639^4 + 18 \, 796 \, 760^4\)
\(\displaystyle 638 \, 523 \, 249^4\) \(=\) \(\displaystyle 630 \, 662 \, 624^4 + 275 \, 156 \, 240^4 + 219 \, 076 \, 465^4\)

$\blacksquare$


Source of Name

This entry was named for Leonhard Paul Euler.


Historical Note

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