Find prime numbers $p,q$ such that: $pq| p^p+q^q+1$

Question

Le $p,q$ be prime numbers such that: $pq| p^p+q^q+1$

Find $p,q$

I don't have any ideas about this problem :(

Thanks :)

Answer 1

I know there is probably a method to solving this other than guessing and checking, but I guessed and checked anyway.

Let $p = 2$ and $q = 5$. Then $pq = 2\cdot 5 = 10$.

$$p^{p} + q^{q} + 1 = 2^{2} + 5^{5} + 1 = 4 + 3125 + 1 = 3130$$

and $3130 \div 10 = 313$, so we do have $$2 \cdot 5 \mid (2^{2} + 5^{5} + 1)$$

Answer 2

If $pq | p^p + q^q + 1$ then also $p | p^p + q^q + 1$. We know that $p|p^p$ so $p$ also has to divide $q^q + 1$. The same we get for $q : q|p^p + 1$. With congruence we can write: $$q^q \equiv -1 \mod p$$ $$p^p \equiv -1 \mod q$$ From Fermat's Little Theorem we have two more congruences: $$q^p \equiv q \mod p$$ $$p^q \equiv p \mod q$$ Now you only have to solve this set of congruences. The only two solutions $(p,q)$ are $(2, 5)$ and $(5, 2)$.