Shanta Laishram

activity

20102026

most citedSome irreducibility results for truncated binomial expansions

10 citations · 42 across the 20 of their papers we have counts for

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26 papers · 1 filter

math.NT2026

Irreducibility of Generalised Laguerre Polynomials for Large Integer Parameters

Drishya T. Das, Saheli Dutta, Shanta Laishram +1

For a positive integer $n$ , we study the irreducibility of the generalised Laguerre polynomials \[ L_n^{(a)}(x) = \frac{1}{n!}\sum_{j=0}^{n} \binom{n}{j} \frac{(n+a)!}{(j+a)!}(-1)^…

math.NT2026

On The Index of Polynomial Compositions over Valued Fields

Anuj Jakhar, Ravi Kalwaniya, Shanta Laishram +1

Determining whether an algebraic number field admits a power integral basis is a classical problem, but it can be difficult for fields defined by polynomial compositions and dynami…

math.NT2025

Primitive prime divisors in the forward orbit of a polynomial

Shanta Laishram, Sudhansu S. Rout, Prabhakar Yadav

For the polynomial $f (z) \in Q [z]$ , we consider the Zsigmondy set $Z (f, 0)$ associated to the numerators of the sequence ${f^{n} (0)}_{n \geq 0}$ . In this paper, w…

math.NT2025

Behaviour of Newton Polygon over polynomial composition

Anuj Jakhar, Shanta Laishram, Kotyada Srinivas +1

In this paper, we study the structure of Newton polygons for compositions of polynomials over the rationals. We establish sufficient conditions under which the successive vertices…

math.NT2024

Generalized Fruit Diophantine equation over number fields

Satyabrat Sahoo, Shanta Laishram

Let $K$ be a number field and $O_{K}$ be the ring of integers of $K$ . In this article, we study the solutions of the generalized fruit Diophantine equation $ax^d-y^2-z^2 +x…

math.NT2023

$p$ -numerical semigroup of generalized Fibonacci triples

Takao Komatsu, Shanta Laishram, Pooja Punyani

For a nonnegative integer $p$ , we give explicit formulas for the $p$ -Frobenius number and the $p$ -genus of generalized Fibonacci numerical semigroups. Here, the $p$ -numerical semig…

math.NT2022

Stability of Certain Higher Degree Polynomials

Shanta Laishram, Ritumoni Sarma, Himanshu Sharma

One of the interesting problems in arithmetic dynamics is to study the stability of polynomials over a field. In this paper, we study the stability of $f (z) = z^{d} + \frac{1}{c}$ for $d…

math.NT2021

Rational solutions to the Variants of Erdős- Selfridge superelliptic curves

Pranabesh Das, Shanta Laishram, N. Saradha +1

For the superelliptic curves of the form $(x + 1) \dots (x + i - 1) (x + i + 1) \dots (x + k) = y^{ℓ}$ with $x, y \in Q$ , $y \neq = 0$ , $k \geq 3$ , $1 \leq i \leq k$ , $ℓ \geq 2,$ a prim…

math.NT2020★ 1 cited

On members of Lucas sequences which are either products of factorials or product of middle binomial coefficients and Catalan numbers

Shanta Laishram

Let ${U_{n}}_{n \geq 0}$ be a Lucas sequence. Then the equation $∣ U_{n} ∣ = m_{1}! m_{2}! \dots m_{k}!$ with $1 < m_{1} \leq m_{2} \leq \dots \leq m_{k}$ implies $n \in {1, 2, 3, 4, 6, 8, 12}$ . Further…

math.NT2020

On members of Lucas sequences which are products of Catalan numbers

Shanta Laishram, Florian Luca, Mark Sias

We show that if ${U_{n}}_{n \geq 0}$ is a Lucas sequence, then the largest $n$ such that $∣ U_{n} ∣ = C_{m_{1}} C_{m_{2}} \dots C_{m_{k}}$ with $1 \leq m_{1} \leq m_{2} \leq \dots \leq m_{k}$ , where $C_m…

math.NT2019

Progress towards a nonintegrality conjecture

Shanta Laishram, Daniel López-Aguayo, Carl Pomerance +1

Given $r \in N$ , define the function $S_{r} : N \to Q$ by $S_{r} (n) = k = 0 \sum n \frac{k}{k + r} (k n)$ . In $2015$ , the secon…

math.NT2019

Irreducibility of extensions of Laguerre Polynomials

Shanta Laishram, Saranya G. Nair, Tarlok Nath Shorey

For integers $a_{0}, a_{1}, \dots, a_{n}$ with $∣ a_{0} a_{n} ∣ = 1$ and either $α = u$ with $1 \leq u \leq 50$ or $α = u + \frac{1}{2}$ with $1 \leq u \leq 45$ , we prove that $ψ_n^{(α)}(x;a_0,a_1,\cdots…

math.NT2019★ 2 cited

Irreducibility and Galois Groups of Generalized Laguerre Polynomials $L_{n}^{(- 1 - n - r)} (x)$

Ankita Jindal, Shanta Laishram, Ritumoni Sarma

We study the algebraic properties of Generalized Laguerre polynomials for negative integral values of a given parameter which is $L_{n}^{(-1-n-r)}(x)= \sum\limits_{j=0}^{n} \binom{…

math.NT2019

On the Galois group of Generalised Laguerre polynomials II

Shanta Laishram, Saranya G. Nair, Tarlok Nath Shorey

For real number $α,$ Generalised Laguerre Polynomials (GLP) is a family of polynomials defined by \begin{align*} L_n^{(α)}(x)=(-1)^n\displaystyle\sum_{j=0}^{n}\binom{n+α}{n-j}\frac…

math.NT2019

On members of Lucas sequences which are products of factorials

Shanta Laishram, Florian Luca, Mark Sias

Here, we show that if ${U_{n}}_{n \geq 0}$ is a Lucas sequence, then the largest $n$ such that $∣ U_{n} ∣ = m_{1}! m_{2}! \dots m_{k}!$ with $1 < m_{1} \leq m_{2} \leq \dots \leq m_{k}$ satisfies $n<3\times 1…

math.NT2016

Sums of the digits in bases 2 and 3

Jean-Marc Deshouillers, Laurent Habsieger, Shanta Laishram +1

Let b $\geq$ 2 be an integer and let s b (n) denote the sum of the digits of the representation of an integer n in base b. For sufficiently large N , one has Card{n $\leq$ N : |s 3 (…

math.NT2016★ 2 cited

Perfect powers in products of terms of elliptic divisibility sequences

Lajos Hajdu, Shanta Laishram, Márton Szikszai

Diophantine problems involving recurrence sequences have a long history and is an actively studied topic within number theory. In this paper, we connect to the field by considering…

math.NT2013★ 1 cited

Grimm's Conjecture and Smooth Numbers

Shanta Laishram, Ram Murty

Let $g (n)$ be the largest positive integer $k$ such that there are distinct primes $p_{i}$ for $1 \leq i \leq k$ so that $p_{i} ∣ n + i$ . This function is related to a celebrated conjecture…

math.NT2013★ 10 cited

Some irreducibility results for truncated binomial expansions

Sudesh K. Khanduja, Ramneek Khassa, Shanta Laishram

For positive integers $n > k$ , let $P_{n, k} (x) = j = 0 \sum k (j n) x^{j}$ be the polynomial obtained by truncating the binomial expansion of $(1 + x)^{n}$ at the $k^{th}…

math.NT2013★ 6 cited

Extensions of Schur's irreducibility results

Shanta Laishram, T. N. Shorey

We prove that the generalised Laguerre polynomials $L_{n}^{(α)} (x)$ with $0\le \al\le 50$ are irreducible except for finitely many pairs $(n, \al)$ and that these exceptions are ne…

math.NT2013★ 7 cited

Irreducibility of generalized Hermite-Laguerre polynomials

Shanta Laishram, T. N. Shorey

For a rational $q = u + \frac{α}{d}$ with $u, α, d\in \ACOBZ$ with $u \geq 0, 1 \leq α < d$ , $g cd (α, d) = 1$ , the \emph{generalized Hermite-Laguerre polynomials $G_{q} (x)$ } are defined by \begin{…

math.NT2013★ 10 cited

Irreducibility of generalized Hermite-Laguerre Polynomials II

Shanta Laishram, T. N. Shorey

In this paper, we show that for each $n \geq 1$ , the generalised Hermite-Laguerre Polynomials $G_{\frac{1}{4}}$ and $G_{\frac{3}{4}}$ are either irreducible or linear polynomial tim…

math.NT2013★ 3 cited

Irreducibility of generalized Hermite-Laguerre Polynomials III

Shanta Laishram, Tarlok Shorey

For a positive integer $n$ and a real number $α$ , the generalized Laguerre polynomials are defined by \begin{align*} L^{(α)}_n(x)=\sum^n_{j=0}\frac{(n+α)(n-1+α)\cdots (j+1+α)(-x)^j…

math.NT2011

Baker's Explicit abc-Conjecture and applications

Shanta Laishram, T. N. Shorey

The conjecture of Masser-Oesterlé, popularly known as $ab c$ -conjecture have many consequences. We use an explicit version due to Baker to solve a number of conjectures.

Irreducibility of Generalised Laguerre Polynomials for Large Integer Parameters

On The Index of Polynomial Compositions over Valued Fields

Primitive prime divisors in the forward orbit of a polynomial

Behaviour of Newton Polygon over polynomial composition

Generalized Fruit Diophantine equation over number fields

p-numerical semigroup of generalized Fibonacci triples

Stability of Certain Higher Degree Polynomials

Rational solutions to the Variants of Erdős- Selfridge superelliptic curves

On members of Lucas sequences which are either products of factorials or product of middle binomial coefficients and Catalan numbers

On members of Lucas sequences which are products of Catalan numbers

Progress towards a nonintegrality conjecture

Irreducibility of extensions of Laguerre Polynomials

Irreducibility and Galois Groups of Generalized Laguerre Polynomials Ln(−1−n−r)​(x)

On the Galois group of Generalised Laguerre polynomials II

On members of Lucas sequences which are products of factorials

Sums of the digits in bases 2 and 3

Perfect powers in products of terms of elliptic divisibility sequences

Grimm's Conjecture and Smooth Numbers

Some irreducibility results for truncated binomial expansions

Extensions of Schur's irreducibility results

Irreducibility of generalized Hermite-Laguerre polynomials

Irreducibility of generalized Hermite-Laguerre Polynomials II

Irreducibility of generalized Hermite-Laguerre Polynomials III

Baker's Explicit abc-Conjecture and applications

$p$ -numerical semigroup of generalized Fibonacci triples

Irreducibility and Galois Groups of Generalized Laguerre Polynomials $L_{n}^{(- 1 - n - r)} (x)$