Ansh Nileshkumar Shah, a Master of Science candidate in the Department of Mathematics and Statistics, will virtually present the Masters Research Project titled On the Diophantine Equation L_n^(k) – L_m^(k) = 2^x 3^y on Monday, June 22, 2026 at 10:00 AM.
The examination committee includes Supervisor Dr. Omar Kihel and Supervisory Committee Member Dr. Yuanlin Li.
Students (both graduate and undergraduate) as well as other members of the Brock Community are invited to attend. A Microsoft Teams link to the meeting can be found here: Join the meeting.
Keywords: Diophantine equations, Generalized Lucas sequences, Linear forms in logarithms, Baker-Davenport reduction method, Transcendental number theory
Abstract: In this project, we study methods from transcendental number theory, particularly the theory of linear forms in logarithms of algebraic numbers. This theory, initiated by the groundbreaking work of Alan Baker in the 1960s, provides powerful explicit lower bounds for linear combinations of logarithms of algebraic numbers. Baker’s fundamental contributions, developed in the mid-1960s, led to major advances in Diophantine analysis and were recognized with the award of the Fields Medal in 1970. Since then, this theory has become one of the central tools for the effective resolution of Diophantine equations. Refinements such as Matveev’s theorem provide explicit bounds that are especially useful in modern applications.
The goal of this work is to apply these methods to the study of a Diophantine equation involving generalized Lucas sequences. More precisely, we investigate the k-generalized Lucas sequence and consider the Diophantine equation L_n^(k) – L_m^(k) = 2^x 3^y where n, m, k, x, and y are nonnegative integers. This problem extends recent work of Kourouma, Rihane and Togbe, who studied the restricted equation L_n^(k) – L_m^(k) = 2 3^y
To achieve this, we combine several tools from Diophantine approximation. First, we apply Matveev’s theorem on linear forms in logarithms to obtain explicit upper bounds for the variables. Because these bounds are typically very large, we then employ reduction techniques based on a variant of the Baker-Davenport method, as developed by Dujella and Petho, together with Legendre’s criterion. These methods allow us to significantly reduce the bounds and ultimately bring the problem into a computationally tractable range.
Our main result provides a comprehensive description of the solutions of the equation under specific parametric constraints. For the range n ≤ k, we provide a complete resolution of the problem, mapping out the full behavior of the solution space. On the other hand, for the range n > k, we fully solve the equation for the specific exponents x = 2, 3, 4, 5, and 6, demonstrating that the remaining variables are strictly bounded and yield a finite set of solutions. This gives a resolution of the equation under the specified conditions and represents a natural extension of previously known results.
