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Using the 5 constants of Euler's identity $ e^{i\pi} + 1 = 0 $ it is possible to include $ \varphi $ into an equation to give an identity containing six constants as follows: $$ e^{\frac{i\pi}{1+\varphi}} + e^{-\frac{i\pi}{1+\varphi}} + e^{\frac{i\pi}{\varphi}} + e^{-\frac{i\pi}{\varphi}} = 0 $$ See article and OEIS Sequence A193537


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An approximation with an accuracy similar to that of $\pi\approx3$ (error<5%) is given by the sixth root of Gelfond's constant, $$e^{\frac{\pi}{6}}\approx \phi$$ with rational term series $$e^{\frac{\pi}{6}}=\sum_{k=0}^{\infty}\frac{\left(e^{\frac{\pi}{2}} - (-1)^k ...


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Because $G_q$ can (almost) be written as the union of some closed intervals with centers $a/q$, where $a$ ranges from $0$ to $q$. After all, if $x = a/q + \epsilon$, with $\epsilon$ some small positive number the inequality reduces to $$q^{1 - \alpha} \ge \|qx\| = |q\epsilon|$$ or $$\epsilon < q^{-\alpha}$$ This shows that $[a/q, a/q + q^{-\alpha}] ...



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