Finding all prime numbers such that $8p^4-3003$ is also prime

Question

Find all the prime numbers $p$ such that $8p^4-3003$ is also a prime

I was trying to solve the previous problem but I haven't been able to. I was thinking about using Fermat's Little Theorem but I'm not sure. Any help is welcome.

Answer 1

Hint:

If $5\nmid p$, then, by Fermat's Little Theorem, $p^4\equiv1\pmod5$, so $8p^4-3003\equiv \,?\pmod5$.