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Sequence with no constant subsequence mod p

Source: Baltic Way 2016, Problem 5

November 5, 2016
number theory

Problem Statement

Let p>3p > 3 be a prime such that p≡3(mod4).p\equiv 3 \pmod 4. Given a positive integer a0a_0 define the sequence a0,a1,…a_0, a_1, \ldots of integers by an=an−12na_n = a^{2^n}_{n-1} for all n=1,2,….n = 1, 2,\ldots. Prove that it is possible to choose a0a_0 such that the subsequence aN,aN+1,aN+2,…a_N , a_{N+1}, a_{N+2}, \ldots is not constant modulo pp for any positive integer N.N.
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