For each $n \in \mathbb{N}$ show that there exists primes $p_1,p_2,\dots ,p_n$ so that $p_i$ is a quadratic residue modulo $p_j$ for each $i \neq j$.
I was given a hint that you can find these primes always in the form $4k+1$ and then I tried to prove this by induction. First I showed that when $n=2$ then you can take $p_1 = 5$ and $p_2 = 29$ and then this holds.
I have been having trouble proving the induction step. I cannot seem to prove that when you have $p_1,p_2\dots,p_{n-1}$ for which is holds then I cannot construct $p_n$ such that this still holds. Any suggestions?