Smallest Fourth Power as Sum and Difference of Fourth Powers

Theorem

The smallest $4$th power that can be expressed as the sum of $2$ $4$th powers minus a $3$rd is:

$2401 = 7^4 = 227^4 + 157^4 - 239^4$

with all numbers less than $10^4$.

Proof

\(\ds \)

\(\)

\(\ds 227^4 + 157^4 - 239^4\)

\(\ds \)

\(=\)

\(\ds 2 \, 655 \, 237 \, 841 + 607 \, 573 \, 201 - 3 \, 262 \, 808 \, 641\)

\(\ds \)

\(=\)

\(\ds 2401\)

\(\ds \)

\(=\)

\(\ds 7^4\)

This theorem requires a proof. In particular: It remains to be shown that this is the smallest such. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• Oct. 1983: Aurel J. Zajta: Solutions of the Diophantine Equation $A^4 + B^4 = C^4 + D^4$ (Math. Comp. Vol. 41, no. 164: pp. 635 – 659)  www.jstor.org/stable/2007700

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $2401$