IR-TST 2017 Problem 4

We arranged all the prime numbers in the ascending order: $p_1=2<p_2<p_3<\cdots$ . Also assume that $n_1<n_2<\cdots$ is a sequence of positive integers that for all $i=1,2,3,\cdots$ the equation $x^{n_i} \equiv 2 \pmod {p_i}$ has a solution for $x$ . Is there always a number $x$ that satisfies all the equations?

Proposed by Mahyar Sefidgaran , Yahya Motevasel