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Satyanand Singh

Publications and source records attributed to Satyanand Singh.

8 recordsLinked to original sources
arXiv2023

Solving the membership problem for certain subgroups of $SL_2(\mathbb{Z})$

For positive integers $u$ and $v$, let $L_u=\begin{bmatrix}1 & 0 \\u&1\end{bmatrix}$ and $R_v=\begin{bmatrix}1 & v \\ 0 & 1\end{bmatrix}$. Let $G_{u,v}$ be the group generated by $L_u$ and $R_v$. In a previous paper, the authors determined a characterization of matrices $M=\begin{bmatrix}a & c \\b&d\end{bmatrix}$ in $G_{u,v}$ when $u,v\geq 3$ in terms of the short continued fraction representation of $b/d$. We extend this result to the case where $u+v> 4$. Additionally, we compute $[\mathscr{G}_{u,v}\colon G_{u,v}]$ for $u,v\geq 1$, extending a result of Chorna, Geller, and Shpilrain.

math.GR↗
arXiv2020

Maximal entries of elements in certain matrix monoids

Let $L_u=\begin{bmatrix}1 & 0\\u & 1\end{bmatrix}$ and $R_v=\begin{bmatrix}1 & v\\0 & 1\end{bmatrix}$ be matrices in $SL_2(\mathbb Z)$ with $u, v\geq 1$. Since the monoid generated by $L_u$ and $R_v$ is free, we can associate a depth to each element based on its product representation. In the cases where $u=v=2$ and $u=v=3$, Bromberg, Shpilrain, and Vdovina determined the depth $n$ matrices containing the maximal entry for each $n\geq 1$. By using ideas from our previous work on $(u,v)$-Calkin-Wilf trees, we extend their results for any $u, v\geq 1$ and in the process we recover the Fibonacci and some Lucas sequences. As a consequence we obtain bounds which guarantee collision resistance on a family of hashing functions based on $L_u$ and $R_v$.

math.NT↗
arXiv2020

Subgroups of $SL_2(\mathbb{Z})$ characterized by certain continued fraction representations

For positive integers $u$ and $v$, let $L_u=\begin{bmatrix} 1 & 0 \\ u & 1 \end{bmatrix}$ and $R_v=\begin{bmatrix} 1 & v \\ 0 & 1 \end{bmatrix}$. Let $S_{u,v}$ be the monoid generated by $L_u$ and $R_v$, and $G_{u,v}$ be the group generated by $L_u$ and $R_v$. In this paper we expand on a characterization of matrices $M=\begin{bmatrix}a & b \\c & d\end{bmatrix}$ in $S_{k,k}$ and $G_{k,k}$ when $k\geq 2$ given by Esbelin and Gutan to $S_{u,v}$ when $u,v\geq 2$ and $G_{u,v}$ when $u,v\geq 3$. We give a simple algorithmic way of determining if $M$ is in $G_{u,v}$ using a recursive function and the short continued fraction representation of $b/d$.

math.GR↗
arXiv2019

Limit points of Nathanson's Lambda sequences

We consider the set $A_{n}=\displaystyle\cup_{j=0}^{\infty}\{\varepsilon_{j}(n)\cdot n^j\colon\varepsilon_{j}(n)\in\{0,\pm1,\pm2,...,\pm\lfloor{{n}/{2}}\rfloor\}\} $. Let $\mathcal{S}_{\mathcal{A}}= \bigcup_{a \in\mathcal{A} } A_{a}$ where $\mathcal{A}\subseteq \mathbb{N}$. We denote by $λ_{\mathcal{A}}(h)$ the smallest positive integer that can be represented as a sum of $h$, and no less than $h$, elements in $\mathcal{S}_{\mathcal{A}}$. Nathanson studied the properties of the $λ_\mathcal{A}(h)$-sequence and posed the problem of finding the values of $λ_\mathcal{A}(h)$. When $\mathcal{A}=\{2,i\}$, we represent $λ_{\mathcal{A}}(h)$ by $λ_{2,i}(h)$. Only the values $λ_{2,3}(1)=1$, $λ_{2,3}(3)=5$, $λ_{2,3}(3)=21$ and $λ_{2,3}(4)=150$ are known. In this paper, we extend this result. For any odd $i>1$ and $h\in\{1,2,3\}$, we find the values of $λ_{2,i}(h)$. Furthermore, for fixed $h\in\{1,2,3\}$, we find the values of $λ_{2,i}(h)$ that occur infinitely many times as $i$ runs over the odd integers bigger than 1. We call these numbers the $\textit{limit points of Nathanson's lambda sequences}$.

math.NT↗
arXiv2018

Mean Row Values in $(u,v)$-Calkin-Wilf Trees

We fix integers $u,v \geq 1$, and consider an infinite binary tree $\mathcal{T}^{(u,v)}(z)$ with a root node whose value is a positive rational number $z$. For every vertex $a/b$, we label the left child as $a/(ua+b)$ and right child as $(a+vb)/b$. The resulting tree is known as the $(u,v)$-Calkin-Wilf tree. As $z$ runs over $[1/u,v]\cap \mathbb{Q}$, the vertex sets of $\mathcal{T}^{(u,v)}(z)$ form a partition of $\mathbb{Q}^+$. When $u=v=1$, the mean row value converges to $3/2$ as the row depth increases. Our goal is to extend this result for any $u,v\geq 1$. We show that, when $z\in [1/u,v]\cap \mathbb{Q}$, the mean row value in $\mathcal{T}^{(u,v)}(z)$ converges to a value close to $v+\log 2/u$ uniformly on $z$.

math.NT↗
arXiv2015

The (u,v)-Calkin-Wilf Forest

In this paper we consider a refinement, due to Nathanson, of the Calkin-Wilf tree. In particular, we study the properties of such trees associated with the matrices $L_u=\begin{bmatrix} 1 & 0 \\ u & 1\end{bmatrix}$ and $R_v=\begin{bmatrix} 1 & v \\ 0& 1\end{bmatrix}$, where $u$ and $v$ are nonnegative integers. We extend several known results of the original Calkin-Wilf tree, including the symmetry, numerator-denominator, and successor formulas, to this new setting. Additionally, we study the ancestry of a rational number appearing in a generalized Calkin-Wilf tree.

math.NT↗
arXiv2013

Perfect Powers of Five with Few Ternary Digits

In this note we will analyze a diophantine equation raised by Michael Bennett in [1] that is pivotal in establishing that powers of five has few digits in its ternary expansion. We will show that the Diophantine equation $3^{a}+3^{b}+2=n^5$, where $(n,3)=1$ and $a>b>0$ is insoluble for pairs of positive integers $(a,b)$ where they are both even or one is even and the other is odd. In the case where both $(a,b)$ are odd, there is one known solution $2^5=3^3+3^1+2.$ We will show that there are no other solutions to the diophantine equation for $n^{5}<32\left(1+3(10^6)\right)^5$.

math.NT↗