How To Build A Special Sequence

Question

I am trying to build a special sequence of positive integers such that the product of any two terms is not a perfect square and also to prove the fact of coprimality of any two terms is not needed.For example the terms of Fermat's numbers are pairwise coprime and using this fact we can prove that the product of any two Fermat's numbers is not a perfect square.So this will not work.So please give me some idea or hint.Thanks in advance.

Answer 1

You can build such a sequence recursively. Start with $1,2$. If you've chosen $a_1, a_2, \ldots, a_n$ then there must be some $N > a_n$ such that none of the products with an already chosen $a_i$ is a square. That must be true since you could always take $N$ to be a prime, but you don't need anything about primes to find $N$. Just start testing with $N = a_n +1, a_n + 2, \ldots$ until you get an $N$ that works. Use that $n$ for $a_{n+1}$.

(This is not an efficient algorithm!)