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finite_geometric_series [2016/06/10 01:35] nikolaj |
finite_geometric_series [2016/06/10 01:42] nikolaj |
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The proof of the infinitude of primes using Fermat numbers uses this. | The proof of the infinitude of primes using Fermat numbers uses this. | ||
- | >Factorization of $a^n-b^n$: | + | In $\mathbb C$, the equation $(x/b)^n=1$ is solved by $x=b\cdot{\mathrm e}^{2\pi i\frac{k}{n}}$, so |
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- | >https://en.wikipedia.org/wiki/Factorization#Sum.2Fdifference_of_two_cubes | + | $a^n-b^n = (a-b)\prod_{k=1}^{n-1} (a-b\cdot{\mathrm e}^{2\pi i\frac{k}{n}})$ |
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- | >Is the following true? How to prove it? | + | |
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- | >$a^n-b^n=\prod_{k=1}^n (a-b\cdot{\mathrm e}^{2\pi i\frac{k}{n}})=(a-b)\prod_{k=1}^{n-1} (a-b\cdot{\mathrm e}^{2\pi i\frac{k}{n}})$ | + | |
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<code> | <code> | ||
n = 6; | n = 6; |