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finite_geometric_series [2016/06/10 01:42]
nikolaj
finite_geometric_series [2016/06/10 02:00] (current)
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.
  
-In $\mathbb C$, the equation $(x/b)^n=1$ is solved by $x=b\cdot{\mathrm e}^{2\pi i\frac{k}{n}}$,​ so+In $\mathbb C$, the equation $\left(\frac{x}{b}\right)^n=1$ is solved by $x=b\cdot{\mathrm e}^{2\pi i\frac{k}{n}}$,​ so
  
 $a^n-b^n = (a-b)\prod_{k=1}^{n-1} (a-b\cdot{\mathrm e}^{2\pi i\frac{k}{n}})$ $a^n-b^n = (a-b)\prod_{k=1}^{n-1} (a-b\cdot{\mathrm e}^{2\pi i\frac{k}{n}})$
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