Solutions of Diophantine Equation x^4 + y^4 = z^2 + 1 for x = 239

Theorem

Consider the indeterminate Diophantine equation:

$x^4 + y^4 = z^2 + 1$

When $x = 239$ and $x > y$, there are $3$ solutions:

\(\ds 239^4 + 104^4\)

\(=\)

\(\ds 58 \, 136^2 + 1\)

\(\ds 239^4 + 143^4\)

\(=\)

\(\ds 60 \, 671^2 + 1\)

\(\ds 239^4 + 208^4\)

\(=\)

\(\ds 71 \, 656^2 + 1\)

Proof

This theorem requires a proof. In particular: It remains to be shown there are no other solutions. This can be achieved by brute force. For this page, it is then adequate to present a pseudocode program to illustrate it. 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

• 1986: Joseph McLean: On Sums of Unlike Powers (Mathematical Spectrum Vol. 18, no. 3: pp. 88 – 89)
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $239$