Omni-MATH / For a given positive integer n and prime number p, find the minimum value of positive integer m that satisfies the following property: for any polynomial …

For a given positive integer n and prime number p, find the minimum value of positive integer m that satisfies the following property: for any polynomial f(x)=(x+a₁)(x+a₂)…(x+a_(n)) (a₁,a₂,…,a_(n) are positive integers), and for any non-negative integer k, there exists a non-negative integer $k'$ such that $$v_p(f(k))<v_p(f(k'))\leq v_p(f(k))+m.$$ Note: for non-zero integer N,v_(p)(N) is the largest non-zero integer t that satisfies p^(t)∣N.

For a given positive integer $n$ and prime number $p$, find the minimum value of positive integer $m$ that satisfies the following property: for any polynomial $$f(x)=(x+a_1)(x+a_2)\ldots(x+a_n)$$ ($a_1,a_2,\ldots,a_n$ are positive integers), and for any non-negative integer $k$, there exists a non-negative integer $k'$ such that $$v_p(f(k))<v_p(f(k'))\leq v_p(f(k))+m.$$ Note: for non-zero integer $N$,$v_p(N)$ is the largest non-zero integer $t$ that satisfies $p^t\mid N$.