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integer root of x^n+a_1x^{n -1}+..+a_n=0 where a_i=product of first i primes

Source: 239 MO 2021 10-11 p1 , 8-9 p2

May 2, 2021
primesIntegerpolynomialInteger Polynomialalgebra

Problem Statement

You are given nn different primes p1,p2,...,pnp_1, p_2,..., p_n. Consider the polynomial xn+a1xn−1+a2xn−2+...+an−1x+anx^n + a_1x^{n -1} + a_2x^{n - 2} + ...+ a_{n - 1}x + a_n, where aia_i is the product of the first ii given prime numbers. For what nn can it have an integer root?
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