MathLabs

Problem 3

Prove that the set of integers of the form 2k−32^k-3 (k=2,3,…k=2,3,\ldots) contains an infinite subset in which every two members are relatively prime.
Step 5 of 5: Conclude the infinite subset
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{An=2an−3:n≥1} is infinite and pairwise coprime\{A_n=2^{a_n}-3:n\ge1\}\text{ is infinite and pairwise coprime}
Detailed analysis

The recursive sequence has infinitely many distinct exponents and every pair of its terms is relatively prime. Since each term has the required form 2k−32^k-3 with k≥2k\ge2, these terms form the desired infinite subset.