arXiv subjects

Paulius Virbalas

Publications and source records attributed to Paulius Virbalas.

2 recordsLinked to original sources

arXiv2026

On the exponential Diophantine equation $(a^n+1)(b^n+1)=x^2$

We study the Diophantine equation $(a^n+1)(b^n+1)=x^2$, which belongs to the family of equations originating from the work of Szalay in 2000. If $a>1$, it is shown that the equation of the title has only one solution in positive integers, when $a$ and $b$ are distinct powers of the same integer $t>1$. Also, a complete description of the solutions is obtained under the assumptions that $a$ and $b$ are coprime and $n$ is even. Several other special cases of the equation are considered, and two conjectures are proposed.

math.NT↗

arXiv2026

Perfect powers among Jacobsthal numbers

Jacobsthal numbers are an example of a Lucas sequence defined by a recurrence relation analogous to that of the Fibonacci numbers, but with different parameters. In this paper, we prove that the only perfect powers among Jacobsthal numbers are the trivial ones, namely $0$ and $1$. Using the Binet formula, the problem is reduced to an exponential Diophantine equation in three unknowns. We resolve this equation via the modular approach, following the framework developed by Bennett and Skinner for ternary Diophantine equations. This work contributes to the study of perfect powers in linear recurrence sequences.

math.NT↗