Proving there are infinitely many integers having the identical set of prime factors.

Question

Let positive integers $a$ and $b$, and let $a_0, a_1, a_2 \ldots$ where $a_i = a + b*i$ is the infinite arithmetic sequence they determine. Prove that there are infinitely many $a_i$ having the identical set of prime factors.

Answer 1

Notice that it is enough to prove that the sequence $x_n=nb+1$ contains infintely many integers which have the same prime factors. In fact, if such a subsequence $x_{n_i}$ exists, the sequence $a+b(an_i)=ax_{n_i}$ is contained in the sequence $a_i$ and those numbers have the same set of prime factors.

Now observe that $x_n^2=n^2b^2+2nb+1=(n^2b+2n)b+1=x_{n^2b+2n}$, and obviously $x_n$ and $x_n^2$ have the same prime factors. Keep squaring to find the required sequence.