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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
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 $$e^{\frac{\pi}{6}}=\sum_{k=0}^{\infty}\frac{\left(e^{\frac{\pi}{2}} - (-1)^k ... 1 Because G_q can (almost) be written as the union of some closed intervals with centers a/q, where a ranges from 0 to q. After all, if x = a/q + \epsilon, with \epsilon some small positive number the inequality reduces to$$q^{1 - \alpha} \ge \|qx\| = |q\epsilon|$$or$$\epsilon < q^{-\alpha} This shows that \$[a/q, a/q + q^{-\alpha}] ...