Can Sum of a Rational Number and Its Reciprocal Be an Integer?

So $fracmnfracnmfracm^2n^2mn$ an integer. This means $$m^2-n^2 mod mn $$ $$m^2n^2kmn$$ for some integer $k$.If $k1$, then $(mn)(m-n)0$ so $mn$ or $m-n$.

With some calculus, it is best shown properties about its zeros, the equation $x^2y^2zxy$. But there are no zeros outside of cases where $m,n0; mn; m-n$

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Can Sum of a Rational Number and Its Reciprocal Be an Integer? 1

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