MathDB

p > 3

is a prime number such that

p|2^{p-1} - 1

and

p \nmid 2^x - 1

for

x = 1, 2,...,p-2

. Let

p = 2k + 3

. Now we define sequence

\{a_n\}

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

2k

consecutive terms of sequence

a_{x+1},a_{x+2},..., a_{x+2k}

such that

a_{x+i } \not\equiv a_{x+j}

(mod

p

) for all

1 \le i < j \le 2k

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