$$\cos(\sqrt{n^2-1}) = \frac{1}{n}$$ I was wondering if there existed integer solutions to to this equation apart from n=1. I've thought that there are probably no more solutions, because RHS is rational and I believe that LHS is not rational in general. I've found theorems about the output of the cosine when the argument is rational, but not for when it is irrational. (Niven's theorem). How could this be solved?
2026-04-08 11:09:13.1775646553
What are the integer solutions to $\cos(\sqrt{n^2-1}) = \frac{1}{n}$?
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It is a consequence of Baker's Theorem, for example as listed here, that if the output of a trigonometric function is algebraic (and so rational numbers like $1/n$ count) then the input is either rational or transcendental. Since $\sqrt{n^2-1}$ is not rational but is algebraic for $n$ apart from 1, there are no other integer solutions to your equation.