3^x + 4^y equals 5^z has Unique Solution/Mistake

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Source Work

1997: David Wells: Curious and Interesting Numbers (2nd ed.):

The Dictionary
$2$


Mistake

Also, the only solution of $3^x + 4^y = 5^z$ in integers is $x = y = z = 2$. The equation $5^x + 12^y = 13^z$ has the same unique solution.


Correction

What it should say is:

Also, the only solution of $3^x + 4^y = 5^z$ in (strictly) positive integers is $x = y = z = 2$ ...

as we also have the solutions:

\(\ds 3^0 + 4^1\) \(=\) \(\ds 5^1\)
\(\ds 5^0 + 12^1\) \(=\) \(\ds 13^1\)

In the original work from which Wells's information was taken, the correct specification for $x$, $y$ and $z$ was used.


This factoid was not included in the first edition of Curious and Interesting Numbers.


Sources