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exchange sequence of natural number

Source: Iranian 3rd round Number Theory exam P2

September 22, 2014
modular arithmeticnumber theory proposednumber theory

Problem Statement

We say two sequence of natural numbers A=(a1,...,ana_1,...,a_n) , B=(b1,...,bnb_1,...,b_n)are the exchange and we write A∼BA\sim B. if 503∣ai−bi503\vert a_i - b_i for all 1≤i≤n1\leq i\leq n. also for natural number rr : ArA^r = (a1r,a2r,...,anra_1^r,a_2^r,...,a_n^r). Prove that there are natural number k,mk,m such that : ii)250≤k250 \leq k
iiii)There are different permutations π1,...,πk\pi _1,...,\pi_k from {1,2,3,...,5021,2,3,...,502} such that for 1≤i≤k−11\leq i \leq k-1 we have πim∼πi+1\pi _i^m\sim \pi _{i+1}
(15 points)
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