Is x^{\sqrt{5}}=y for Rational Numbers?

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The discussion centers on the equation x^{\sqrt{5}}=y for rational numbers x and y. It concludes that the only rational solution is the trivial case where x=y=1. The Gelfond–Schneider theorem is cited as the definitive proof that no other rational pairs exist for this equation. The algorithm mentioned aims to compute logarithms in phinary base without float multiplications, but the impossibility of finding additional rational solutions limits its application.

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I've programmed an algorithm to numerically compute the logarithm of numbers in phinary base easily. I could avoid float multiplications if I can find a pair of rational numbers x and y such that
x^{\sqrt{5}}=y
Is it possible?
Probably not, but I cannot prove it :(
 
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No, it isn't possible except for the obvious x=y=1. There's a http://en.wikipedia.org/wiki/Gelfond–Schneider_theorem" which says that it's impossible.
 
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