An Application of Fermat's Last Theorem

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Can you find all integers n such that n+9 and n^2+27 are both cube numbers? The solution to this problem uses Fermat's Last Theorem in a really cool way exhibiting that Fermat's Last Theorem has real world applications.

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I can pronounce the formula for the proof of Fermat's Great Theorem:
1 - Fermat's Great Theorem NEVER! and nobody! NOT! HAS BEEN PROVEN !
2 - proven! THE ONLY POSSIBLE proof of Fermat's Great Theorem !
3 - Fermat's Great Theorem is proved universally-proven for all numbers !
4 - Fermat's Great Theorem is proven in the requirements of himself! Fermata 1637 y.
5 - Fermat's Great Theorem proved in 2 pages of a notebook !
6 - Fermat's Great Theorem is proved in the apparatus of Diophantus arithmetic !
7 - The proof of the great Fermat's Great Theorem, as well as the formulation,
is easy for a student of the 5th grade of the school to understand !!!
8 - Me! opened the GREAT! A GREAT Mystery! Fermat's Great Theorem !
(not a "simple" "mechanical" proof

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