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 1 of 5: Start the recursive construction
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a1=3,An=2an−3,A1=5a_1=3,\qquad A_n=2^{a_n}-3,\qquad A_1=5
Detailed analysis

Set a1=3a_1=3 and define An=2an−3A_n=2^{a_n}-3. Then A1=23−3=5A_1=2^3-3=5. We will recursively choose larger exponents so that each new AnA_n is coprime to all previous ones.