arXiv subjects

S. S. Rout

Publications and source records attributed to S. S. Rout.

12 recordsLinked to original sources

arXiv2026

The quotient problem for linear recurrence sequences

Let $\{U(m)\}_{m\in \N}$ and $\{V(n)\}_{n\in \N}$ be linear recurrence sequences. It is a well-known Diophantine problem to determine the finiteness of the set of natural numbers $n$ such that the ratio $U(n)/V(n)$ is an integer. We study the finiteness problem for the set $(m, n)\in \mathbb{N}^2$ such that there exist non-zero positive integers $d_{m, n}$ satisfying $\log |d_{m, n}|=o(n)$, and $d_{m, n}U(m)/V(n)$ is an element from a finitely generated subring of $\C$. In particular, we prove that for $m\neq n $, there exists a polynomial $P$ such that $d_{m, n}P(n)U(m)/V(n)$ is a multi-recurrence and $V(n)/P(n)$ is a linear recurrence and for $m=n$ both $d_{m, n}P(n)U(m)/V(n)$ and $V(n)/P(n)$ are linear recurrences. To prove our results, we employ Schmidt's subspace theorem, and the concept of moving hyperplanes, moving polynomials, and moving points.

math.NT↗

arXiv2026

Uniform bounds on $S$-integral points in backward orbits

Let $K$ be a number field with algebraic closure $\overline{K}$ and let $S$ be a finite set of places of $K$ containing all the archimedean places. It is known from Silverman's result that a forward orbit of a rational map $φ$ contains finitely many $S$-integers in the number field K when $φ^2$ is not a polynomial. Sookdeo stated an analogous conjecture for the backward orbits of a rational map $φ$ using a general $S$-integrality notion based on the Galois conjugates of points. He proved his conjecture for the power map $φ(z) =z^d$ for $d \geq 2$ and consequently for Chebyshev maps (J. Number Theory 131 (2011), 1229-1239). In this paper, we establish uniform bounds on the number of $S$-integral points in the backward orbits of any non-zero $β$ in $K$, relative to a non-preperiodic point $α\in \mathbb{P}^1(\overline{K})$, under the power map $φ(z) =z^d $.

math.NT↗

arXiv2025

Algebraic approximations to linear combinations of S-units

Let $Γ\subset \bar{\Q}^{\times}$ be a finitely generated multiplicative group of algebraic numbers, let $α_1,\ldots,α_m$ be non-zero algebraic numbers, and let $\varepsilon >0$ be fixed. In this paper, we prove that there exist only finitely many tuples $(u_1, \ldots, u_m, q, p)\in Γ^m\times\mathbb{Z}^2$ with $d = [\mathbb{Q}(u_1, \ldots, u_m):\mathbb{Q}]$ such that for any two tuples $(u_1,\ldots,u_m)$ and $(u'_1,\ldots,u'_m)$, we have $\frac{u_{i_1}}{u_{i_2}}\neq \frac{u'_{i_1}}{u'_{i_2}}$ for $1\leq i_1\neq i_2\leq m$ and it is stable under Galois conjugation over $\Q$, $\max\{|α_1 qu_1|, \ldots, |α_m qu_m|\}>1$, the tuple $(α_1qu_1, \ldots, α_mq u_m)$ is not pseudo-Pisot and \[0< \left|\sum_{i=1}^m α_iq u_i - p\right|<\frac{1}{\left(\prod_{i=1}^mH( u_i)\right)^{\varepsilon} |q|^{md+\varepsilon}},\] where $H(u_i)$ denotes the absolute Weil height. This result extends one of the main results of Corvaja-Zannier \cite{corv}. In addition, we prove a result similar to \cite[Theorem 1.4]{kul} in a more general setting. In our proofs, we exploit the subspace theorem based on the work of Corvaja-Zannier.

math.NT↗

arXiv2025

Zeros and $S$-units in sums of terms of recurrence sequences in function fields

Let $(U_n)_{n\geq 0}$ be a non-degenerate linear recurrence sequence with order at least two defined over a function field and $\mathcal{O}_S^*$ be the set of $S$-units. In this paper, we use a result of Brownawell and Masser to prove effective results related to the Diophantine equations concerning linear recurrence sequences and $S$-units. In particular, we provide a finiteness result for the solutions of the Diophantine equation $U_{n_1} + \cdots + U_{n_r} \in \mathcal{O}_S^*$ in nonnegative integers $n_1, \ldots, n_r$. Furthermore, we study the finiteness result of the Diophantine equation $U_n+V_m+W_\ell = 0$ in $(n, m, \ell)\in \N^3$, where $U_n,V_m,W_\ell$ are simple linear recurrence sequences in the function field.

math.NT↗

arXiv2024

Sum of terms of recurrence sequences and $S$-units in the solution sets of norm form equations

In this paper, we prove two results related to the solutions of norm form equations. Firstly, we give a finiteness result for sums of terms of linear recurrence sequences appearing in the coordinates of solutions of norm form equations. Next, we give a finiteness result concerning solutions of norm form equations representable as sums of $S$-units with a fixed number of terms. To prove these results, we use a deep results concerning the finiteness of the solutions of polynomial-exponential equations and $S$-unit equations.

math.NT↗

arXiv2024

On the resolution of the Diophantine equation $U_n + U_m = x^q$

Suppose that $(U_{n})_{n \geq 0}$ is a binary recurrence sequence and has a dominant root $α$ with $α>1$ and the discriminant $D$ is square-free. In this paper, we study the Diophantine equation $U_n + U_m = x^q$ in integers $n \geq m \geq 0$, $x \geq 2$, and $q \geq 2$. Firstly, we show that there are only finitely many of them for a fixed $x$ using linear forms in logarithms. Secondly, we show that there are only finitely many solutions in $(n, m, x, q)$ with $q, x\geq 2$ under the assumption of the {\em abc-conjecture}. To prove this, we use several classical results like Schmidt subspace theorem, a fundamental theorem on linear equations in $S$-units and Siegel's theorem concerning the finiteness of the number of solutions of a hyperelliptic equation.

math.NT↗

arXiv2023

Sums of $S$-units in sum of terms of recurrence sequences

Let $S := \{p_1,\ldots ,p_{\ell}\}$ be a finite set of primes and denote by $\mathcal{U}_S$ the set of all rational integers whose prime factors are all in $S$. Let $(U_n)_{n\geq 0}$ be a non-degenerate linear recurrence sequence with order at least two. In this paper, we provide a finiteness result for the solutions of the Diophantine equation $aU_n + bU_m = z_1 +\cdots +z_r,$ where $n\geq m$ and $z_1, \ldots, z_r\in \mathcal{U}_S$.

math.NT↗

arXiv2020

$S$-parts of sums of terms of linear recurrence sequences

Let $S= \{ p_1, \ldots, p_s\}$ be a finite, non-empty set of distinct prime numbers and $(U_{n})_{n \geq 0}$ be a linear recurrence sequence of integers of order $r$. For any positive integer $k,$ we define $(U_j^{(k)})_{j\geq 1}$ an increasing sequence composed of integers of the form $U_{n_k} +\cdots + U_{n_1}, \ n_k>\cdots >n_1$. Under certain assumptions, we prove that for any $ε>0,$ there exists an integer $n_{0}$ such that $[U_j^{(k)}]_S < \left(U_j^{(k)}\right)^ε,$ for $ j > n_0,$ where $[m]_S$ denote the $S$-part of the positive integer $m$. On further assumptions on $(U_{n})_{n \geq 0},$ we also compute an effective bound for $[U_j^{(k)}]_S$ of the form $\left(U_j^{(k)}\right)^{1-c}$, where $c $ is a positive constant depends only on $(U_{n})_{n \geq 0}$ and $S.$

math.NT↗

arXiv2017

Prime powers in sums of terms of binary recurrence sequences

Let $\{u_{n}\}_{n \geq 0}$ be a non-degenerate binary recurrence sequence with positive, square-free discriminant and $p$ be a fixed prime number. In this paper, we have shown the finiteness result for the solutions of the Diophantine equation $u_{n_{1}} + u_{n_{2}} + \cdots + u_{n_{t}} = p^{z}$ with some conditions on $n_i $ for all $1\leq i \leq t$. Moreover, we explicitly find all the powers of three which are sums of three balancing numbers using the lower bounds for linear forms in logarithms. Further, we use a variant of Baker-Davenport reduction method in Diophantine approximation due to Dujella and Pethő.

math.NT↗

arXiv2017

Perfect powers in alternating sum of consecutive cubes

In this paper, we consider the problem about finding out perfect powers in an alternating sum of consecutive cubes. More precisely, we completely solve the Diophantine equation $(x+1)^3 - (x+2)^3 + \cdots - (x + 2d)^3 + (x + 2d + 1)^3 = z^p$, where $p$ is prime and $x,d,z$ are integers with $1 \leq d \leq 50$.

math.NT↗

arXiv2016

Linear combinations of prime powers in sums of terms of binary recurrence sequences

Let $\{ {U_{n}\}_{n \geq 0} }$ be a non-degenerate binary recurrence sequence with positive discriminant. Let $\{p_1,\ldots, p_s\}$ be fixed prime numbers and $\{b_1,\ldots ,b_s\}$ be fixed non-negative integers. In this paper, we obtain the finiteness result for the solution of the Diophantine equation $U_{n_{1}} + \cdots + U_{n_{t}} = b_1 p_1^{z_1} + \cdots+ b_s p_s^{z_s} $ under certain assumptions. Moreover, we explicitly solve the equation $F_{n_1}+ F_{n_2}= 2^{z_1} +3^{z_2}$, in non-negative integers $n_1, n_2, z_1, z_2$ with $z_2\geq z_1$. The main tools used in this work are the lower bound for linear forms in logarithms and the Baker-Davenport reduction method.

math.NT↗

arXiv2013

Affine Kac-Moody symmetric spaces associated with untwisted Kac-Moody algebras

In this paper we have computed all the affine Kac-Moody symmetric spaces which are tame Frechet manifolds starting from the Vogan diagrams related to the affine untwisted Kac-Moody algebras. The detail computation of affine Kac-Moody symmetric spaces associated with A_1^(1) and A_2^(1) are shown algebraically to corroborate our method.

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