MathDB

Prove that there exist two infinite sequences

\{a_n\}_{n\ge 1}

and

\{b_n\}_{n\ge 1}

of positive integers such that the following conditions hold simultaneously:

(i)

0 < a_1 < a_2 < a_3 < \cdots

;

(ii)

a_n < b_n < a_n^2

, for all

n\ge 1

;

(iii)

a_n \minus{} 1 divides b_n \minus{} 1, for all

n\ge 1

(iv)

a_n^2 \minus{} 1 divides b_n^2 \minus{} 1, for all

n\ge 1

[19 points out of 100 for the 6 problems]

Problem Statement