We continue the study from [1], by studying equations of type $\psi(n) = \dfrac{k+1}{k} \cdot \ n+a,$ $a\in \{0, 1, 2, 3\},$ and $\varphi(n) \dfrac{k-1}{k} n-a,$ 3\}$ for $k > 1,$ where $\psi(n)$ $\varphi(n)$ denote Dedekind, respectively Euler's, arithmetical functions.

A bstract In this work, we propose a string-inspired two fields inflation model to address the fine-tuning problem that standard suffers. The fast-rolling tachyon $$ \mathcal{T} T originated from D-brane and anti-D-brane pair annihilation locks inflaton φ slowly rolling on Higgs-like potential V\left(\varphi ...

Let \(\theta \) be an irrational number and \(\varphi a real number. \(C > 2\) \(\varepsilon 0\). There are infinitely many positive integers n free of prime factors \(> (\log )^C\), such that \(\Vert \theta + \varphi \Vert < n^{-\left( \frac{1}{3} - \frac{2}{3C}\right) \varepsilon }\). Here y\Vert is the distance from y to \(\mathbb Z\).