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finite_geometric_series [2016/06/10 00:38] nikolaj |
finite_geometric_series [2016/06/10 01:35] 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. | ||
| - | prove this, maybe by induction | + | >Factorization of $a^n-b^n$: |
| - | + | > | |
| - | $a^n-b^n=\prod_{k=1}^n (a-b\cdot{\mathrm e}^{2\pi i\frac{k}{n}})=(a-b)\prod_{k=1}^{n-1} (a-b\cdot{\mathrm e}^{2\pi i\frac{k}{n}})$ | + | >https://en.wikipedia.org/wiki/Factorization#Sum.2Fdifference_of_two_cubes |
| + | > | ||
| + | >Is the following true? How to prove it? | ||
| + | > | ||
| + | >$a^n-b^n=\prod_{k=1}^n (a-b\cdot{\mathrm e}^{2\pi i\frac{k}{n}})=(a-b)\prod_{k=1}^{n-1} (a-b\cdot{\mathrm e}^{2\pi i\frac{k}{n}})$ | ||
| + | > | ||
| <code> | <code> | ||
| n = 6; | n = 6; | ||
