Problem 3

Prove that the set of integers of the form

2^k-3

(

k=2,3,\ldots

) contains an infinite subset in which every two members are relatively prime.

Step 4 of 5: Prove the new term is coprime to all old terms

Previous step Next step

p\mid A_n\text{ and }p\mid A_k\implies p\mid(A_n+2)-A_n=2

Detailed analysis

Suppose a prime $p$ divides both $A_n$ and an earlier $A_k$ . By the previous step, $p$ also divides $A_n+2$ . Hence $p$ divides their difference $2$ . But $p$ is an odd divisor of the odd number $A_k$ , impossible. Therefore $\gcd(A_n,A_k)=1$ for every $k<n$ .