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Proposition: Prop. 9.17: Last Element of Geometric Progression with Coprime Extremes has no Integer Proportional as First to Second
(Proposition 17 from Book 9 of Euclid's “Elements”)
If any multitude whatsoever of numbers is in continued proportion, and the outermost of them are prime to one another, then as the first (is) to the second, so the last will not be to some other (number).
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Proofs: 1
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BYSA 3.0 Sources:
 Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016