brianchung11 wrote: Prove that there are infinitely many primes $ p$ such that the total number of solutions mod $ p$ to the equation $ 3x^{3} + 4y^{4} + 5z^{3} - y^{4}z \equiv 0$ is $ p^2$ If p is a prime in the form 6k+5 then $ \{x^3|x = 0,...,p - 1\}$ is a complete reside mod p . It is obvious because the equation $ x^3\equiv 1 (\mod p)$ has only solution $ x\equiv 1 (\mod p)$ .Because $ \gcd(p,3) = 1$ ,so the set $ \{3x^3|x = 0,...,p - 1\}$ is also a complete reside mod p . Therefore for each pairs of $ (z_0,y_0)$ there exist only $ x_0$ such that $ p|3x^3 + 4y_{0}^4 + 5z_{0}^3 - y_{0}^4z_{0}$ There are exactly $ p^2$ pairs of $ (z,y)$ chose from the set $ \{0,..,p - 1\}$ , so the equation mod p also has $ p^2$ solution . This proof completes .