Prime factor $>250000$ for $1002004008016032$

Question

I need to find a prime factor, $p$, of $1002004008016032$ such that $p \gt 250000$. Now this is a very large number and I know it would be stupid of me to factorize such a large number into its prime factors . Any help how I can solve this ?

Answer 1

$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+x^{n-3}y^2+\cdots+xy^{n-2}+y^{n-1})$

Taking $x=10^3,y=2$, we get

$1002004008016032=x^5+x^4y+x^3y^2+x^2y^3+xy^4+y^5=\frac{x^6-y^6}{x-y}$

Also,

$\frac{x^6-y^6}{x-y} = (x+y)(x^2+xy+y^2)(x^2-xy+y^2)$

=$1002\times1002004\times998004 = 1002\times16\times250501\times249501$

Hence, the prime number , $p\gt250000$ is $250501$