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 2 of 5: Choose the next exponent

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a_n=a_{n-1}\prod_{p\mid A_{n-1}}(p-1)

Detailed analysis

For each $n>1$ , let the product range over all distinct prime divisors $p$ of $A_{n-1}$ , and define $a_n=a_{n-1}\prod_{p\mid A_{n-1}}(p-1)$ . Every $A_{n-1}$ is odd, so these primes are odd; the construction also makes the sequence strictly increasing from $a_1=3$ .