### Distribution of difference between inverses of consecutive integers modulo $p$.

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We consider a conjecture of Erdős and Rosenfeld and a conjecture of Ruzsa when the number is a perfect square. In particular, we show that every perfect square n can have at most five divisors between $\surd n-\u221cn{\left(logn\right)}^{1/7}$ and $\surd n+\u221cn{\left(logn\right)}^{1/7}$.

We obtain an asymptotic formula for the number of visible points (x,y), that is, with gcd(x,y) = 1, which lie in the box [1,U] × [1,V] and also belong to the exponential modular curves $y\equiv a{g}^{x}\left(modp\right)$. Among other tools, some recent results of additive combinatorics due to J. Bourgain and M. Z. Garaev play a crucial role in our argument.

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