For $$\frac{x^3-y^3}{x-y}=x^2+xy+y^2=p$$ $p=6k+1$ give $p$ prime.
On what conditions the diophantine equation $$\frac{x^5-y^5}{x-y}=p$$ is solvable in integers. Does it have a linear expression for $$\frac{x^n-y^n}{x-y}=p$$n =7,11,13...etc Is it easy to solve and in what conditions it's solvable in $\mathbb Z[i]$.
Write out $$ x^4 + x^3 y + x^2 y^2 + x y^3 + y^4. $$ Up to some large positive $N,$ the maximum number of possible values of this for $0 < y < x$ is no larger than $\sqrt N.$ As a result, it is not possible to describe the set of primes just by congruences. The quantitative aspect of Dirichlet's result on primes in arithmetic progressions is this: given fixed $a,b > 0,$ with $\gcd(a,b)=1,$ the number of primes in the progression $a t + b,$ up to $N,$ is asymptotically $$ \frac{N}{\phi(a) \log N.} $$ This is much, much larger than $\sqrt N.$