Euler 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:

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

$\blacksquare$


Also known as

The Euler Quartic Conjecture is also known as Euler's conjecture.

However, there is more than one such conjecture, so $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers the longer form to remove ambiguity as to which is being referenced.


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