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Consecutive terms of sequence

Source: Korea National 2012 Problem 5

August 19, 2012
inductionmodular arithmeticnumber theory proposednumber theory

Problem Statement

p>3 p >3 is a prime number such that p∣2p−1−1 p | 2^{p-1} -1 and p∤2x−1 p \not | 2^x - 1 for x=1,2,⋯ ,p−2 x = 1, 2, \cdots , p-2 . Let p=2k+3 p = 2k+3 . Now we define sequence {an} \{ a_n \} as ai=ai+k=2i(1≤i≤k), aj+2k=ajaj+k (j≥1) a_i = a_{i+k}= 2^i ( 1 \le i \le k ) , \ a_{j+2k} = a_j a_{j+k} \ ( j \ge 1 ) Prove that there exist 2k2k consecutive terms of sequence ax+1,ax+2,⋯ ,ax+2k a_{x+1} , a_{x+2} , \cdots , a_{x+2k} such that for all 1≤i<j≤2k 1 \le i < j \le 2k , ax+i≢ax+j (mod p) a_{x+i} \not \equiv a_{x+j} \ (mod \ p) .
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