arXiv subjects

Pritam Kumar Bhoi

Publications and source records attributed to Pritam Kumar Bhoi.

7 recordsLinked to original sources

arXiv2026

Some Diophantine Equations involving associated Pell numbers and repdigits

In this paper, we explore the relationship between repdigits and associated Pell numbers, specifically focusing on two main aspects: expressing repdigits as the difference of two associated Pell numbers, and identifying which associated Pell numbers can be represented as the difference of two repdigits. Additionally, we investigate all associated Pell numbers which are the concatenation of three repdigits. Our proof utilizes Baker's theory on linear forms in logarithms of algebraic numbers, along with the Baker-Davenport reduction technique. The computations were carried out with the help of a simple computer program in {\it Mathematica}.

math.GM↗

arXiv2026

Resolution of the Skolem Problem for $k$-Generalized Lucas Sequences

This paper provides a complete solution to Skolem's problem for the $k$-generalized Lucas sequence $(L_n^{(k)})_{n \in \mathbb{Z}}$ with a primary focus on its behavior at negative indices. We characterize the zero-distribution of this sequence by identifying and bounding all indices $n < 0$ such that $L_n^{(k)} = 0$. Our central result establishes that the zero-multiplicity $δ_k$ of the sequence is $(k-1)(k-2)/2$ for all $k.$

math.NT↗

arXiv2026

Solving Skolem problem for negative indexed $k-$generalized Pell numbers

In this paper, we address the Skolem problem for the $k$-generalized Pell sequence $(P_n^{(k)})_{n\geq2-k}$ extended to negative indices. We focus on identifying and bounding the indices $n<0$ for which $P_n^{(k)}=0.$ In particular, we establish that the zero multiplicity of $P_n^{(k)}$ is $ χ_k = \lfloor k^2/4\rfloor$ for all $k \in [4, 500].$

math.NT↗

arXiv2024

Balancing and Lucas-balancing numbers as difference of two repdigits

Positive integers with all digits equal are called repdigits. In this paper, we find all balancing and Lucas-balancing numbers, which can be expressed as the difference of two repdigits. The method of proof involves the application of Baker's theory for linear forms in logarithms of algebraic numbers and the Baker-Davenport reduction procedure.

math.NT↗

arXiv2024

Sum of terms of recurrence sequences in the solution sets of generalized Pell equations

Let $(X_{k})_{k\geq 1}$ and $(Y_k)_{k\geq 1}$ be the sequence of $X$ and $Y$-coordinates of the positive integer solutions $(x, y)$ of the equation $x^2 - dy^2 = t$. In this paper we completely describe those recurrence sequences such that sums of two terms recurrence sequences in the solution sets of generalized Pell equations are infinitely many. Further, we give an upper bound for the number of such terms when there are only finitely many of them. This work is motivated by the recent paper Hajdu and Sebestyén (Int. J. Number Theory 18 (2022), 1605-1612).

math.NT↗

arXiv2024

Repdigits as difference of two balancing or Lucas-balancing numbers

Repdigits are natural numbers formed by the repetition of a single digit. In this paper, we study the problem of writing repdigits as the difference of two balancing or Lucas-balancing numbers. The method of proof involves the application of Baker's theory for linear forms in logarithms of algebraic numbers and the Baker-Davenport reduction procedure. Computations are done with the help of a simple computer program in {\it Mathematica}.

math.GM↗

arXiv2022

On perfect powers that are sum of two balancing numbers

Let $B_k$ denote the $k^{th}$ term of balancing sequence. In this paper we find all positive integer solutions of the Diophantine equation $B_n+B_m = x^q$ in variables $(m, n,x,q)$ under the assumption $n\equiv m \pmod 2$. Furthermore, we study the Diophantine equation \[B_n^{3}\pm B_m^{3} = x^q\] with positive integer $q\geq 3$ and $\gcd(B_n, B_m) =1$.

math.NT↗