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number theory with sequence, related to prime p=2k+3

Source: KJMO 2012 p6

May 4, 2019
number theorySequenceprime

Problem Statement

p>3p > 3 is a prime number such that p∣2p−1−1p|2^{p-1} - 1 and p∤2x−1p \nmid 2^x - 1 for x=1,2,...,p−2x = 1, 2,...,p-2. Let p=2k+3p = 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_ja_{j+k} \,\, (j \le 1) Prove that there exist 2k2k consecutive terms of sequence ax+1,ax+2,...,ax+2ka_{x+1},a_{x+2},..., a_{x+2k} such that ax+i≢ax+ja_{x+i } \not\equiv a_{x+j} (mod pp) for all 1≤i<j≤2k1 \le i < j \le 2k .
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