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Periodic sequence

Source: ITAMO 2021 Problem 6

June 3, 2021
number theoryprime numbers

Problem Statement

A sequence x1,x2,...,xn,...x_1, x_2, ..., x_n, ... consists of an initial block of pp positive distinct integers that then repeat periodically. This means that {x1,x2,…,xp}\{x_1, x_2, \dots, x_p\} are pp distinct positive integers and xn+p=xnx_{n+p}=x_n for every positive integer nn. The terms of the sequence are not known and the goal is to find the period pp. To do this, at each move it possible to reveal the value of a term of the sequence at your choice.
(a) Knowing that 1≤p≤101 \le p \le 10, find the least nn such that there is a strategy which allows to find pp revealing at most nn terms of the sequence. (b) Knowing that pp is one of the first kk prime numbers, find for which values of kk there exist a strategy that allows to find pp revealing at most 55 terms of the sequence.
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