In [5], we give a sufficient condition for generalized continued fractions to converge to an irrational number. Specifically, we determine how many non-integers must exist in $c_{1},c_{2},\ldots, c_{i}$ in order for the continued fraction $[0; c_{1},c_{2},\ldots]$ to be irrational, where $c_{i}$ are rational numbers greater than $M>1$ (denoted by $c_{i}\in \mathbb{Q}_{\ge M}$) and the denominator of $c_{i}$ less than or equal to positive integer $N$. In [6], we extended this result from the case $M>1$ to $M>0$. In this paper, we give a further extension by removing the restrictions on the denominators of the partial quotients. Let $M$ be a real number greater than $0$. We consider continued fractions $[0;c_{1},c_{2},\ldots]$, where $c_{i}$ are rational numbers greater than or equal to $M$ for any positive integer $i$. We give a sufficient condition for it to converge to an irrational number and some interesting examples.
In the paper, a new version of the binary/strong Goldbach’s conjecture is considered and discussed. It is based on quadratic forms representation and on a new conjecture, that establishes equivalence.
This paper is devoted to the study of the recurrence function of q-ary Thue–Morse word $t_q$. After giving the structure of singular factors, we describe the maximal return time of factors of $t_q$. We conclude by determining the recurrence function of generalized Thue–Morse word.
We obtain refinements and new results related to an extension of Dedekind’s arithmetic function, introduced in [1].
Schinzel's classification of radical extensions relies on the structure of quadratic cyclotomic extensions, which also play an important role in the study of cyclotomic extensions over general fields. In this paper, we identify properties and invariants associated with quadratic cyclotomic extensions that are relevant to these classification problems. We determine the possible minimal polynomials of primitive roots of unity generating quadratic extensions, describe the prime-power levels at which such extensions occur, and assemble the prime-power information into a global classification of quadratic cyclotomic fields. The treatment applies in arbitrary characteristic, with the usual restriction that, in positive characteristic, the orders of the roots of unity considered are prime to the characteristic.
