Let $\psi:\mathbb{Z}_{\geq 1}\rightarrow$ $\mathbb{Z}_{\geq 1}$ be increasing.We study the arithmetical properties of the values of the C-fraction \[\xi:=1+\frac{c_{1}q^{\psi\left( 1\right) }}{1}%%TCIMACRO{\QATOP{{}}{+}}%%BeginExpansion\genfrac{}{}{0pt}{}{{}}{+}%%EndExpansion\frac{c_{2}q^{\psi\left( 2\right) }}{1}%%TCIMACRO{\QATOP{{}}{+\cdots+}}%%BeginExpansion\genfrac{}{}{0pt}{}{{}}{+\cdots+}%%EndExpansion\frac{c_{n}q^{\psi\left( n\right) }}{1}%%TCIMACRO{\QATOP{{}}{+\cdots}}%%BeginExpansion\genfrac{}{}{0pt}{}{{}}{+\cdots}%%EndExpansion\]where $\left( c_{n}\right) _{n\geq1}$ and $q$ are non zero rational numbers satisfying suitable conditions. Under these conditions, we prove that $\xi$ is irrational and give lower and upper bounds for its irrationality exponent. Inthe most favorable cases, we deduce from these bounds and Roth's theorem that $\xi$ is transcendental.
Let $K$ be a $p$-adic field, namely a finite extension of the field of $p$-adic numbers $\mathbb{Q}_p$. If $G$ is a finitely generated subgroup of $K^\times$, we describe how to compute the degrees of the Kummer extensions $K(\zeta_{n}, \sqrt[n]G)/K(\zeta_n)$ for all positive integers $n$. Moreover, we compare a~Kummer extension of number fields with the corresponding Kummer extensions of $p$-adic fields which arise by completion.
We prove an exact formula for the expected density of distribution of complex level crossings of random polynomials spanned by orthogonal polynomials on the unit circle and unit disk. We then inquire into the consequences of the asymptotical evaluations.
A new method for solving quartic equations due to Luo and Lin is investigated both computationally and theoretically. As a result, a completely straightforward elementary method is given for solving Bumby's equation 3X^4-2Y^2=1, along with a conjecture, which if resolved, would enable a similar proof for a possibly infinite family of similar equations.
We prove that the sequence of the last nonzero digits of factorials in every integer base b>2 is not eventually periodic. We also extend the Adamczewski–Bugeaud criterion, originally formulated for integer base expansions, to Cantor base expansions associated with a periodic Cantor base. As an application, we show that a certain real number expressed through a Cantor base expansion is transcendental when the Cantor base and the digit sequence satisfy suitable conditions.
