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

Theorem

The Diophantine equation:

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

has exactly one solution in (strictly) positive integers:

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

Proof

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Sources

• 1956: L. Jeśmanowicz: Kilka uwag o liczbach pitagorejskich (Wiadom. Mat. Ser. 2 Vol. 1: pp. 196 – 202)

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

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