Solutions of a cubic diophantine equation in $\mathbb{Z}/p\mathbb{Z}$ Suppose $p\in\mathbb{Z}$ is prime and $p\equiv 1\pmod{3}$. Is there an estimate of the number of  solutions of $x^3+y^3=z^3$ in $\mathbb{Z}/p\mathbb{Z}$, preferably using elementary number theory and algebra ? Can something be said about the "expected" ( in the sense of Probability Theory ) number of solutions ? I am particularly interested in large values of $p$ and elementary techniques.
 A: This particular problem is discussed in detail in Chapter 7 of Rational Points on Elliptic Curves by Silverman and Tate. 
Let $p$ be a prime number. Let $M_p$ denote the number of (projective) solutions to $x^3+y^3\equiv z^3 (\text{mod } p)$ in $\mathbb{Z}/p\mathbb{Z}$. Gauss proved that if $p\equiv 1 (\text{mod } 3)$, then there are integers $A$, and $B$ such that $4p=A^2+27B^2$. Up to sign, $A$ and $B$ are unique. Furthermore, if we fix the sign of $A$ to be positive, then in fact $M_p = p+1+A$. 
So indeed, the theorem tells us exactly the number of solutions in $\mathbb{Z}/p\mathbb{Z}$ for this elliptic curve! I find this result striking and beautiful.
The proof given in the book is somewhat long, and uses clever manipulation of Gauss sums. 
Remark. Last line of Jyrki's comment relates to quite an interesting result called Hasse's bound. Note that in the above case, this bound is verified: indeed, $4p=A^2+27B^2$ implies $A^2\le 4p$, giving us $A\le 2\sqrt{p}$. I highly recommend the above book by Silverman and Tate for accessible introduction to this topic and beyond.
