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Abstract

Let $f_{1},\ldots ,f_{N}$ be arithmetic functions, i.e., functions whose domain is equal to $\mathbb {N}$. Suppose that for any $j=1,\ldots ,N$ there exist $a_{j,1}\ldots ,a_{j,n_{j}}\in \mathbb {N}$ and $b_{j,1},\ldots ,b_{j,n_{j}}\in \mathbb {Z}\setminus \{0\}$ such that $f_{j}(p)=(a_{j,1}p+b_{j,1})\cdot \ldots \cdot (a_{j,n_{j}}p+b_{j,n_{j}})$ for any prime $p$ and that $f_{j}(p_{1}\ldots p_{k})=f_{j}(p_{1})\ldots f_{j}(p_{k})$ for any distinct primes $p_{1},\ldots ,p_{k}$. We prove that for any positive integer $r$ there exist infinitely many positive integers $m$ such that all numbers $f_{j}(m)$, $j=1,\ldots ,N$, are $r$th powers of integers.