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For every positive integer $n$ let $p\left(n\right)$ be the largest prime number $p\le n$. Given a positive integer $n=n_1$, we study the positive integer $r=R\left(n\right)$ such that if we define recursively $n_{i+1}=n_i-p\left(n_i\right)$ for $i\ge 1$, then $n_r$ is a prime or $1$. We obtain upper bounds for $R\left(n\right)$ as well as an estimate for the set of $n$ whose $R\left(n\right)$ takes on a fixed value $k$.

In this article we study, using elementary and combinatorial methods, the distribution of quadratic non-residues which are not primitive roots modulo $p^h$ or $2p^h$ for an odd prime $p$ and $h\ge 1$ an integer.