<s>[INST] How many solutions to prime = $a^3+b^3+c^3 - 3abc$ Let $a,b,c$ be integers.
Let $p$ be a given prime. 
How to find the number of solutions to $p = a^3+b^3+c^3 - 3abc$ ?
Another question is ; let $w$ be a positive integer. Let $f(w)$ be the number of primes of type prime = $a^3+b^3+c^3 - 3abc$ below $w$. How does the function $f(w)$ behave ? How fast does it grow ? Are those primes of type $A$ $mod$ $B$ for some integers $A$ and $B$ ? 
How to deal with this ?
Can this be solved without computing the class number ?
 [/INST] We may also prove that there are infinitely many primes when $a+b+c=1$.  This answer along with Andre Nicolas' answer and Will Jagy's answer allows us to calculate the exact number of  representations $a^3+b^3+c^3-abc=p$ for each prime $p$.
We begin with the factorization $$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca).$$  When $a+b+c=1$, we have that $$a^2+b^2+c^2-ab-bc-ca=3a^2+3ab+3b^2-3a-3b+1.$$  Rearranging, this becomes $$\frac{3}{4}\left(2a+b-1\right)^2+\frac{1}{4}\left(3b-1\right)^2.$$  If $x=a+2b-1,$ and $y=a-b,$ then the above is $$\frac{3}{4}\left(x+y\right)^{2}+\frac{1}{4}\left(x-y\right)^{2}=x^{2}+xy+y^{2}.$$ This is the norm form for $\mathbb{Z}\left[\zeta_{3}\right],$ the Eisenstein integers, and we know exactly which primes $p$ can be represented as $$p=x^{2}+xy+y^{2}.$$ To count the number of such representations, see my answer here. Note that this happens if and only if $p\equiv1 \pmod{6}$, and so we see that when $a+b+c=1$, the polynomial $a^3+b^3+c^3-3abc$ represents precisely those primes which are $1$ modulo $6$. 
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