Related papers: On the representation of an integer in Ostrowski a…
We will provide algorithmic implementation with proofs of existence and uniqueness for the Absolute and Alternating Ostrowski Numeration Systems.
Let $\alpha$ be a fixed quadratic irrational. Consider the Diophantine equation \[ y^a\ =\ q_{N_1} + \cdots + q_{N_K},\quad N_1 \geq \cdots \geq N_{K} \geq 0,\quad a, y \geq 2 \] where $(q_N)_{N\,\geq\,0}$ is the sequence of convergent…
In this article, we propose a variant of the usual Ostrowski $\alpha$-numeration (where $\alpha$ is a real in [0, 1[) that codes integers (positive as well as negative) and reals of [0, 1[ (instead of [--$\alpha$, 1--$\alpha$[), so that for…
Given a negative $D>-(\log X)^{\log 2-\delta}$, we give a new upper bound on the number of square free integers $<X$ which are represented by some but not all forms of the genus of a primitive positive definite binary quadratic form $f$ of…
We study the effect of addition on the Hamming weight of a positive integer. Consider the first $2^n$ positive integers, and fix an $\alpha$ among them. We show that if the binary representation of $\alpha$ consists of $\Theta(n)$ blocks of…
Let $(\alpha,\mathcal{N}_{\alpha})$ and $(\beta,\mathcal{N}_{\beta})$ be two canonical number systems for an imaginary quadratic number field $K$ such that $\alpha$ and $\beta$ are multiplicatively independent. We provide an effective lower…
We present an elementary three pass algorithm for computing addition in Ostrowski numeration systems. When $a$ is quadratic, addition in the Ostrowski numeration system based on $a$ is recognizable by a finite automaton. We deduce that a…
In a multi-base representation of an integer (in contrast to, for example, the binary or decimal representation) the base (or radix) is replaced by products of powers of single bases. The resulting numeral system has desirable properties…
We use recurrence equations (alias difference equations) to enumerate the number of formula-representations of positive integers using only addition and multiplication, and using addition, multiplication, and exponentiation, where all the…
For two distinct integers $m_1,m_2\ge2$, we set $\alpha_1=[0;\overline{1,m_1}]$ and $\alpha_2=[0;\overline{1,m_2}]$ and we denote by $S_{\alpha_1}(n)$ and $S_{\alpha_2}(n)$ respectively the sum of digits functions in the Ostrowski…
Recently the author used certain quaternion orders to demonstrate the universality of some quaternary quadratic forms. Here a further study is done on one of these orders analogous to Hurwitz's proof of the formula for the number of…
We provide algorithms for the absolute and alternating Ostrowski Expansions of the continuum and provide proofs for their uniqueness.
Using a method we have utilized previously, namely through a finite power series expansion which also sometimes is known as the "radix polynomial" representation of an integer, we find an upper bound for a van der Waerden number that has a…
In this paper we consider certain quaternary quadratic forms and octonary quadratic forms and by using the theory of modular forms, we find formulae for the number of representations of a positive integer by these quadratic forms.
This paper is the continuation of \cite{htl}, where we deal with Lucas sequences. Here we study integers represented by integer sequences which satisfy binary recursive relations. In case of non-degenerate sequences we give bounds for the…
Recently, the authors showed that for every irrational number $\alpha$, there exist infinitely many positive integers $n$ represented by any given positive definite binary quadratic form $Q$, satisfying $||\alpha n||<n^{-(1/2-\varepsilon)}$…
We prove an upper bound for the number of representations of a positive integer $N$ as the sum of four $k$-th powers of integers of size at most $B$, using a new version of the Determinant method developed by Heath-Brown, along with recent…
We construct a Beurling generalized number system satisfying the Riemann hypothesis and whose integer counting function displays extremal oscillation in the following sense. The prime counting function of this number system satisfies…
A sharp upper bound for the maximum integer not belonging to an ideal of a numerical semigroup is given and the ideals attaining this bound are characterized. Then the result is used, through the so-called Feng-Rao numbers, to bound the…
We present the first linear lower bound for the number of bits required to be accessed in the worst case to increment an integer in an arbitrary space- optimal binary representation. The best previously known lower bound was logarithmic. It…