Find all natural numbers $n$ such that $2n+1$ and $3n+1$ are square numbers and $2n+9$ is a prime.

Question

Find all natural numbers $n$ such that $2n+1$ and $3n+1$ are square numbers and $2n+9$ is a prime.

---

I can prove:

$n$ divide by $8$ leaves $0$; $n$ divide by $5$ leaves $0$

So $n$ divide by $40$ leaves $0$ and let $n=40k (k\in N^*)$

And i knew the answer is $n=40\Leftrightarrow k=1$ but i can not how to find $k=1$. Help me, please.

Answer 1

Hint: We have a description of the integers $n$ such that $2n+1$ and $3n+1$ are perfect squares, see

Positive integer $n$ such that $2n+1$ , $3n+1$ are both perfect squares

There are answers referring to Putnam competition where very similar questions have been solved. This, and the above link, might be helpful.