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# Prime factorization of a huge number

Hi!

Does anyone know how to solve this problem:

Find one prime factor of $$1+{ 2 }^{ 21 }+{ 4 }^{ 21 }$$.

Thanks for any help in advance!

Note by Julian Yu
1 year, 8 months ago

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Let $$x=2^3$$. We get the expression as $$x^{14}+x^7+1$$.

Note that $$\omega$$ and $$\omega^2$$ divides this expression where $$\omega,\omega^2$$ are the complex cube roots of unity.

The expression which has its roots as the complex cube roots of unity is $$x^2+x+1$$.

Therefore $$x^2+x+1$$ divides $$x^{14}+x^7+1$$ or $$2^6+2^3+1=73$$ divides $$1+2^{21}+4^{21}$$.

- 1 year, 8 months ago