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Number Theory from Iranian TST 2017

Source: Iranian TST 2017, first exam day 2, problem 4

April 6, 2017
number theoryIranIranian TSTprime numbersmodular arithmetic

Problem Statement

We arranged all the prime numbers in the ascending order: p1=2<p2<p3<⋯p_1=2<p_2<p_3<\cdots. Also assume that n1<n2<⋯n_1<n_2<\cdots is a sequence of positive integers that for all i=1,2,3,⋯i=1,2,3,\cdots the equation xni≡2(modpi)x^{n_i} \equiv 2 \pmod {p_i} has a solution for xx. Is there always a number xx that satisfies all the equations?
Proposed by Mahyar Sefidgaran , Yahya Motevasel
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