Thabit and Williams Numbers Base $b$ as a Sum or Difference of Two $g$-Repdigits

We investigate cases where Thabit and Williams numbers in base $b$ can be expressed as the sum or difference of two $g$-repdigits. For specific values of $b$ and $g$, we describe parametric solutions yielding infinitely many solutions for some equations and establish upper bounds for the parameters of the remaining finitely many solutions. As an illustration, we also provide a complete solution for some equations.

💡 Research Summary

**
The paper investigates when Thabit numbers and Williams numbers expressed in base b can be written as the sum or difference of two g‑repdigits (numbers whose base‑g representation consists of a single repeated digit). A Thabit number of the first kind in base b has the form ((b+1)b^{n}-1), the second kind ((b+1)b^{n}+1); a Williams number in base b is ((b-1)b^{n}\pm1). A g‑repdigit is (\displaystyle \frac{a(g^{m}-1)}{g-1}) with (1\le a\le g-1) and (m\ge1). The central Diophantine problems are

\