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finite_geometric_series [2016/06/10 00:38] nikolaj |
finite_geometric_series [2016/06/10 01:41] 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 | + | In $\mathbb C$, the equation $(x/b)^n=1$ is solved by $x=b\cdot{\mathrm e}^{2\pi i\frac{k}{n}$, so |
| $a^n-b^n=\prod_{k=1}^n (a-b\cdot{\mathrm e}^{2\pi i\frac{k}{n}})=(a-b)\prod_{k=1}^{n-1} (a-b\cdot{\mathrm e}^{2\pi i\frac{k}{n}})$ | $a^n-b^n=\prod_{k=1}^n (a-b\cdot{\mathrm e}^{2\pi i\frac{k}{n}})=(a-b)\prod_{k=1}^{n-1} (a-b\cdot{\mathrm e}^{2\pi i\frac{k}{n}})$ | ||
