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

This paper investigates the conditions under which Thabit and Williams numbers in base $b$ can be expressed as the sum or difference of two $g$ -repdigits, providing parametric solutions for infinite cases, upper bounds for finite cases, and complete solutions for specific equations.

Original authors: Kouèssi Norbert Adédji, Marija Bliznac Trebješanin, Jelena Pleština

Published 2026-02-10

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Kouèssi Norbert Adédji, Marija Bliznac Trebješanin, Jelena Pleština

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a professional locksmith, but instead of physical keys, you are working with mathematical patterns.

This paper is essentially a high-level investigation into whether two very specific types of "number patterns" can be combined—either by adding them together or subtracting them—to create a third, very specific pattern.

Here is the breakdown of the "characters" in this mathematical story:

1. The "Thabit and Williams" Numbers (The Target Patterns)

Think of these as "The Growing Towers."
Imagine you are building towers out of blocks. A Thabit or Williams number is a tower that grows in a very predictable, explosive way. Every time you add a new level, the tower doesn't just get taller; it multiplies in size based on a specific rule (using a base number $b$ ). They are like a series of expanding ripples in a pond—each one is much larger than the last, following a strict geometric rhythm.

2. The "Repdigits" (The Building Blocks)

Think of these as "The Monotone Chords."
In music, a monotone chord is one where every note is exactly the same. In math, a "repdigit" is a number made of only one repeating digit. For example, in our everyday base-10 system, $777$ or $22,222$ are repdigits. They are "boring" numbers because they lack variety; they are just the same digit repeated over and over.

3. The Big Question: "Can we build a Tower using only two Chords?"

The mathematicians are asking: "Can you take one of those massive, exploding 'Towers' and perfectly construct it by either adding or subtracting exactly two 'Monotone Chords'?"

For example: Can a massive Thabit Tower be equal to $777 + 22$ ? Or Can it be $999 - 11$?

How they solved it: The "Mathematical Microscope"

The researchers didn't just guess and check. They used two heavy-duty tools:

The "Linear Forms in Logarithms" (The Precision Scale): Imagine trying to weigh a single grain of sand on a scale meant for elephants. This mathematical tool allows them to measure the tiny, microscopic gaps between these massive numbers. If the gap is too small or doesn't fit a certain pattern, they can prove a solution is impossible.
The "Reduction Method" (The Sieve): This is like having a giant sieve. They start with a massive range of possibilities (numbers so large they would fill the universe) and use logic to shake the sieve, catching the "impossible" numbers and letting them fall through, until only a tiny handful of potential answers remain.

The Results: What did they find?

The paper reaches three main conclusions:

The "Infinite Exceptions": In a few very specific, "lucky" scenarios (like when the base numbers are perfectly synchronized), there are infinitely many ways to make this work. It’s like finding a special frequency where two different instruments always play the same note.
The "Finiteness Rule": In almost every other case, the answer is "No, or at most, a very small number of times." They proved that for most combinations, you can't keep building these patterns forever; eventually, the "Towers" grow too fast and the "Chords" can't keep up.
The "Complete Map": For the numbers we use every day (Base 10), they actually finished the job. They didn't just say "there are a few solutions"; they found exactly how many there are and what they look like. They provided a complete "map" of every single time these patterns collide.

Summary in one sentence:

The researchers proved that while these two types of mathematical patterns occasionally "shake hands" in perfect harmony, they are mostly strangers that almost never match up.

This paper, titled "Thabit and Williams numbers base $b$ as a sum or difference of two $g$ -repdigits," investigates the intersection of two specific classes of integers: generalized Thabit/Williams numbers and repdigits in a given base $g$ .

1. The Problem

The authors seek to solve specific Diophantine equations that represent the following mathematical questions:

Thabit/Williams Numbers (Base $b$ ): These are generalizations of the classical Thabit ibn Qurra numbers. A Thabit number base $b$ takes the form $(b \pm 1)b^{n \pm 1}$ , and a Williams number base $b$ takes the form $(b \pm 1)b^{n \pm 1}$ .
$g$ -Repdigits: An integer is a $g$ -repdigit if all its digits in base $g$ are identical. Mathematically, a $g$ -repdigit $N$ is expressed as $N = d \frac{g^m - 1}{g - 1}$ , where $d \in \{1, \dots, g-1\}$ .

The core objective is to find all integer solutions $(d_1, d_2, l, m, n)$ for the equations:

$(b \pm 1)b^{n \pm 1} = d_1 \left(\frac{g^l - 1}{g - 1}\right) + d_2 \left(\frac{g^m - 1}{g - 1}\right)$ (Sum of two repdigits)
$(b \pm 1)b^{n \pm 1} = d_1 \left(\frac{g^l - 1}{g - 1}\right) - d_2 \left(\frac{g^m - 1}{g - 1}\right)$ (Difference of two repdigits)

2. Methodology

The paper employs a rigorous number-theoretic approach involving several advanced techniques:

Linear Forms in Logarithms: The authors use Baker’s theory (specifically Lemma 5, a version of Matveev’s theorem) to derive effective upper bounds for the exponents $n, l,$ and $m$ . This transforms an infinite search space into a finite one.
Reduction Method: Because the initial upper bounds provided by Baker's theory are astronomically large (on the order of $10^{29}$ ), the authors apply the Baker-Davenport reduction method (using continued fractions and Lemma 7) to significantly lower these bounds to a computationally manageable range.
Case Analysis: The authors distinguish between cases where the equations yield infinitely many solutions (parametric solutions) and cases where they yield only finitely many solutions.
Computational Verification: For specific practical cases (e.g., $g=10$ and $2 \le b \le 12$ ), the authors use Wolfram Mathematica to implement the reduction algorithms and exhaustively check all remaining candidates.

3. Key Contributions and Results

The paper provides a complete classification of the solutions:

Infinite Families (Proposition 1 & 2): The authors identify specific relationships between $b$ and $g$ (such as $b=2$ and $g=2^k$ ) where the equations possess infinitely many solutions. They provide the explicit parametric forms for these solutions.
Finiteness Theorems (Theorem 3 & 4): They prove that, outside of the infinite families, the number of solutions for both the sum and difference equations is always finite. They provide explicit (though large) theoretical upper bounds for the indices $n, l,$ and $m$ .
Complete Numerical Solutions (Theorem 13 & 19):
- For the sum of two decimal repdigits ( $g=10$ ) and $2 \le b \le 12$ , they prove there are exactly 164 solutions with $n \ge 1$ .
- For the difference of two decimal repdigits ( $g=10$ ) and $2 \le b \le 12$ , they prove there are exactly 160 solutions with $n \ge 1$ .
- They provide specific examples of these representations (e.g., expressing a Thabit number as a combination of decimal repdigits).

4. Significance

This research contributes to the broader field of Diophantine equations and arithmetic properties of number sequences. While Thabit and Williams numbers are traditionally studied for their primality, this paper shifts the focus to their additive/subtractive structure in different bases.

The methodology serves as a robust template for solving similar problems involving other sequences (like Fibonacci or Lucas numbers) and repdigits. By providing both theoretical bounds and exhaustive numerical lists, the paper bridges the gap between abstract number theory and computational mathematics.