# 5^x + 12^y equals 13^z has Unique Solution

From ProofWiki

## Theorem

The Diophantine equation:

- $5^x + 12^y = 13^z$

has exactly one solution in integers:

- $5^2 + 12^2 = 13^2$

## Proof

## Sources

- 1993: Nobuhiro Terai:
*The Diophantine equation $x^2 + q^m = p^n$*(*Acta Arithmetica***Vol. 63**: 351 – 358)

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