# Euler's Quartic Conjecture

## 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

- Oct. 1988: Noam D. Elkies:
*On $A^4 + B^4 + C^4 = D^4$*(*Math. Comp.***Vol. 51**,*no. 184*: 825 – 835) www.jstor.org/stable/2008781

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $20,615,673$

- Weisstein, Eric W. "Diophantine Equation--4th Powers." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/DiophantineEquation4thPowers.html

- Weisstein, Eric W. "Euler Quartic Conjecture." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/EulerQuarticConjecture.html