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I'm trying to understand why pianos "can't be tuned" and am looking for a proof of a corollary of the rational roots theorem found here (looking for proof not by contradiction): https://youtu.be/1Hqm0dYKUx4?t=171

It's stated:

For any integers $a, b$ and $n$ with $n > 1$ then $\big(\frac{a}{b}\big)^n \neq 2$

chankonabe
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  • Also: https://math.stackexchange.com/q/1477179/42969, https://math.stackexchange.com/q/1451172/42969, https://math.stackexchange.com/q/1451172/42969  – all found with Approach0 – Martin R Oct 24 '19 at 07:37
  • Thanks for finding those links. Any chance of a not proof by contradiction? – chankonabe Oct 24 '19 at 07:41
  • @chankonabe Use the same proofs to show that an integer power $\ge 1$ of a non-integer fraction remains a non-integer (so never $= 2)$ – Bill Dubuque Oct 24 '19 at 17:06

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