Existence of Simultaneously Satisfying Sequences of Positive Integers

[ritwik06]

Homework Statement

Prove that there exists two infinite sequences <an> and <bn> of
positive integers such that the following conditions hold simultaneously:
i) 1<a1<a2<a3...;
ii) an<bn<(an)^2 for all n>=1
iii)(an) - 1 divides (bn) - 1 for all n>=1
iv)(an)^2 -1 divides (bn)^2 - 1 for all n>=1

Homework Equations

The Attempt at a Solution

What I guess from this question is that both the series must be odd. Am I
right?

From iii) an iv) I deduce that bn+1 |mod| an+1 =0 for n>=1

Please help me further!

 

[ritwik06]

I am very sorry I keyed in the wrong title for the post! MODS if you could please change it to Sequences?