MathDB

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

Problem Statement