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Why do all primes $n>3$ satisfy $\,309\mid 20^n-13^n-7^n$

Solve the following... $309|(20^n-13^n-7^n)$ in $\mathbb{Z}^+$. I invested lotof time to it and finally went to WolframAlpha for help by typing... Solve $309k=20^n-13^n-7^n$ over the integers. It returned the following...enter image description here enter image description here Note that all the primes are generated here. Can someone please explain why? Thanks! EDIT: the formulas suggested in the answerare too complicated, but this is so simple!



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