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Arithmetic progressions of integers that are relatively prime to their digital sums

This paper investigates the maximum possible lengths of arithmetic progressions consisting of positive integers that are relatively prime to the sum of their digits in a given base $b$ .

Original authors: Ryan Blau, Joshua Harrington, Sarah Lohrey, Eliel Sosis, Tony W. H. Wong

Published 2026-02-10

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Ryan Blau, Joshua Harrington, Sarah Lohrey, Eliel Sosis, Tony W. H. Wong

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

The "Socially Awkward" Numbers: A Story of Math and Mismatches

Imagine you are at a massive party. At this party, there are two groups of people: the "Niven" group and the "Anti-Niven" group.

To figure out which group you belong to, you have to look at two things about yourself:

Your ID Number (your actual value).
The Sum of Your Digits (a little number derived from your ID).

The Niven People are the "perfect matches." Their ID number is perfectly divisible by the sum of their digits. If your ID is 12, your digit sum is $1+2=3$ . Since 12 can be divided by 3, you are a Niven! They are harmonious and orderly.

The Anti-Niven People (the focus of this paper) are the "mismatches." They are relatively prime to their digit sums. This means they share no common factors other than 1. If your ID is 13, your digit sum is $1+3=4$ . Since 13 and 4 share no common factors, you are an Anti-Niven. You are the rebels; you don't "fit" into the patterns of your own digits.

What is this paper actually studying?

The mathematicians in this paper are looking for "Socially Awkward Streaks."

In math, an Arithmetic Progression (AP) is just a sequence of numbers that follows a steady rhythm—like counting by 2s ($2, 4, 6, 8...$) or by 5s ($5, 10, 15, 20...$).

The researchers wanted to know: How long can a streak of "mismatched" (Anti-Niven) numbers last before a "perfect match" (Niven) inevitably breaks the chain?

Think of it like walking down a street of houses. Most houses have a specific pattern. The researchers are asking: "How many houses in a row can I walk past where the house number and its digit sum refuse to cooperate?"

The Big Discoveries

Here is what the researchers found, broken down into simple concepts:

1. The "No Infinite Streaks" Rule

The paper proves that you can never have an infinite streak of Anti-Niven numbers. No matter how you set your rhythm (your "step size"), eventually, the math will force a "perfect match" (a Niven number) to appear. The universe of numbers eventually demands order.

2. The "Smallest Prime" Speed Bump

They discovered that the length of these streaks is often limited by the "smallest prime factor" of the base you are using.

Analogy: Imagine you are walking in a rhythm of 3 steps. If there is a "trap" every 5 steps, you can't walk more than 4 steps without hitting a trap. The math works similarly; certain prime numbers act like "speed bumps" that break the Anti-Niven streak.

3. The "Even vs. Odd" Personality Split

The paper shows that the rules change depending on whether your number base (like our standard Base-10) is even or odd.

When the base is even, the streaks behave in a somewhat predictable, constrained way.
When the base is odd, the streaks can actually be longer and more "rebellious."

4. The "Extreme Streaks"

They found specific, rare scenarios where these streaks can get surprisingly long. For example, in Base-2 (the language of computers), they proved you can find a streak of 5 "mismatched" numbers in a row, and this happens infinitely often throughout the number line.

Summary for the Non-Mathematician

If you think of numbers as a giant, infinite parade, most people are marching in perfect rhythm with their own digits (Niven numbers). This paper is a deep dive into the rebels (Anti-Niven numbers).

The authors have mapped out exactly how long these rebels can march in a row before the "rhythm of the universe" forces them to sync up with their digits. They've proven that while the rebels can form long, impressive lines, they can never march forever. Eventually, the pattern always returns.

This technical summary provides an overview of the research paper "Arithmetic Progressions of Integers that are Relatively Prime to their Digital Sums" by Blau et al. (2023).

1. Problem Statement

The paper investigates the properties of $b$ -anti-Niven numbers.

Definition: An integer $n$ is $b$ -anti-Niven if it is relatively prime to the sum of its digits in base $b$ (i.e., $\gcd(n, s_b(n)) = 1$ ). This is the logical inverse of a $b$ -Niven number, where $s_b(n)$ divides $n$ .
Core Objective: The authors seek to determine the existence, maximum possible length, and frequency of arithmetic progressions (APs) consisting entirely of $b$ -anti-Niven numbers. Specifically, they examine $d$ -APs (arithmetic progressions with common difference $d$ ) of varying lengths and types.

2. Methodology

The authors employ a combination of number theoretic proofs, modular arithmetic, and constructive methods:

Modular Analysis: They use properties of digit sums, specifically the lemma that if $\delta \mid (b-1)$ , then $\delta \mid n \iff \delta \mid s_b(n)$ , to establish constraints on when an AP can contain anti-Niven numbers.
Constructive Proofs: To prove that certain lengths are attainable, they construct specific sequences using large integers $m$ and specific common differences $d$ that force the digit sum to remain constant or follow a predictable pattern.
Density and Prime Distribution: They utilize results related to the distribution of primes and Euler's Theorem to ensure that the constructed terms in an AP do not share common factors with their digit sums.

3. Key Contributions and Results

A. Existence and Finiteness

Infinite APs: The authors prove that no $b$ -anti-Niven $d$ -AP of infinite length exists (Theorem 2.3). This stands in contrast to $b$ -Niven numbers, where infinite $d$ -APs can contain infinitely many terms.
Arbitrary Lengths: They prove that for any integer $t$ , there exist $n$ and $d$ such that a $b$ -anti-Niven $d$ -AP of length $t$ exists (Theorem 2.4).

B. Maximum Lengths of Specific APs

1-APs (Consecutive Integers): For $b > 2$ , let $p$ be the smallest prime dividing $b-1$ . The maximum length of a sequence of consecutive $b$ -anti-Niven numbers is exactly $p-1$ (Theorem 3.2).
2-APs (Even/Odd Steps):
- If $b$ is even and not of the form $2r+1$ , the maximum length is $p-1$ (where $p$ is the smallest odd prime dividing $b-1$ ).
- If $b$ is odd, they establish a lower bound of length $b$ (Theorem 4.1).
$(b-1)$ -APs: For even $b$ , the maximum length is $2b+1$ (Theorem 3.5). They also provide a lower bound for odd primes $b$ of $2b+1$ (Theorem 4.2).

C. Upper Bounds for Odd $d$

The paper identifies cases where the "natural" bound (based on the smallest prime $p \mid b-1$ ) is not reached. For even $b \ge 6$ and odd $3 \le d \le b/2$ , the maximum length is bounded by $\lceil 2b/d \rceil + 2$ (Theorem 3.4).

4. Summary of Results Table

Type of AP	Condition	Result/Bound
Infinite $d$ -AP	Any $b, d$	Length is always finite
1-AP (Consecutive)	$b > 2$	Max length = $p-1$ (where $p$ is smallest prime $\mid b-1$ )
2-AP	$b$ even, $b \neq 2r+1$	Max length = $p-1$ (where $p$ is smallest odd prime $\mid b-1$ )
$(b-1)$ -AP	$b$ even	Max length = $2b+1$
$d$ -AP (odd $d$ )	$b \ge 6$ even, $3 \le d \le b/2$	Max length $\le \lceil 2b/d \rceil + 2$

5. Significance

This paper contributes to the field of additive number theory and the study of digital properties of integers. While much previous literature has focused on Niven (Harshad) numbers, this work shifts the focus to the "anti-Niven" property, providing a complete characterization of the constraints imposed by base- $b$ digit sums on the structure of arithmetic progressions. The results establish a rigorous boundary between the behavior of numbers that are divisible by their digit sums and those that are coprime to them.