Is 312500051 prime?

$312500051 = (50)^{5} + 50 + 1.$

Let $50 = x$ .

Now,

$x^3 \equiv {1}\pmod {x^2 + x + 1}$

$\Rightarrow$ ${x^5 + x + 1}\equiv {x^3.x^2 +x + 1}\equiv {x^2 + x + 1}\equiv {0}\pmod{x^2 + x + 1}$

Therefore $x^2 + x + 1 | x^7 + x + 1$ .

$\Rightarrow$ $50^2 + 50 + 1 | 50^5 + 50 + 1$

$\Rightarrow$ $2551 | 312500051$ .

Therefore $\boxed{2551}$ is the answer.

312500051=2551x122501 therefore the answer is 2551