Prove that the square root of a Mersenne number ($k={2}^{n}-1$) is irrational.

Jaqueline Kirby
2022-06-06
Answered

Prove that the square root of a Mersenne number ($k={2}^{n}-1$) is irrational.

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iceniessyoy

Answered 2022-06-07
Author has **27** answers

All perfect squares are either $1$ or $0\phantom{\rule{0.667em}{0ex}}\mathrm{mod}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}4$. Writing ${2}^{n}-1$ in binary, we get $\mathrm{1111...11}$, and when you divide that by $4$, which is using only the two right most digits (the rest of the digits are a multiple of $100$), it's clear that all Mersenne numbers are $3\phantom{\rule{0.667em}{0ex}}\mathrm{mod}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}4$, except when $n=0,1$.

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