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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 ?

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    $\begingroup$ $2^5\times 3^2\times 7\times 109^2\times 167\times 250501$ $\endgroup$ – polfosol Dec 31 '16 at 18:03
  • $\begingroup$ How did you figure out such nasty factors :) $\endgroup$ – user366398 Jan 1 '17 at 0:25
  • $\begingroup$ Go here $\longrightarrow$ alpertron.com.ar/ECM.HTM It is a calculator and strictly works for integers. When a decimal is put, it will round the result to the nearest integer. When given an equation or inequality, it will return $-1$ if true and $0$ if false. It also has many other functions you can learn :) $\endgroup$ – Feeds Mar 30 '18 at 18:26
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$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+x^{n-3}y^2+\cdots+xy^{n-2}+y^{n-1})$

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 $250501$

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