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Permutation--invariant Niven numbers

This paper introduces the concept of permutation-invariant Niven numbers, proves the existence of infinitely many such numbers with unbounded magnitude, and presents an exhaustive search method for identifying them.

Original authors: Hui-Ling Wu, S. Y. Lou

Published 2026-02-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hui-Ling Wu, S. Y. Lou

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 have a set of numbered tiles, like the ones on a Scrabble board or a combination lock. In the world of mathematics, there's a special club called Niven Numbers (also known as Harshad numbers). To join this club, a number has to be "nice" to its own digits: if you add up all the digits in the number, the original number must be perfectly divisible by that sum.

For example, take the number 12.

Add the digits: $1 + 2 = 3$ .
Is 12 divisible by 3? Yes! ( $12 \div 3 = 4$ ).
So, 12 is a Niven number.

Now, imagine you take those same tiles and start shuffling them around. If you swap the 1 and the 2, you get 21.

Add the digits: $2 + 1 = 3$ .
Is 21 divisible by 3? Yes! ( $21 \div 3 = 7$ ).
So, 21 is also a Niven number.

The New Discovery: The "Unshakeable" Numbers

The paper you shared introduces a new, stricter category of these numbers called Permutation-Invariant Niven Numbers (PINNs).

Think of a PINN as a super-stable building.

A normal Niven number is like a house that stands firm on its foundation.
A PINN is like a house built with magic bricks. No matter how you rearrange the bricks (the digits)—even if you flip the house upside down or swap the front door with the back window—the house still stands perfectly.

In math terms: A PINN is a number where every single possible way you can rearrange its digits (ignoring leading zeros, like turning "012" into "12") results in a number that is still a Niven number.

What Did the Authors Find?

The authors, Huiling Wu and Senyue Lou, went on a treasure hunt to find these "magic numbers." Here is what they discovered, explained simply:

1. They are everywhere (but rare)
They proved that there are infinitely many of these numbers. You can keep finding them forever. However, they are very rare. If you look at the first billion numbers, almost none of them are PINNs. They are the "unicorns" of the number world.

2. The "Zero" Trick
One of the coolest tricks they found involves the number 0.
Imagine you have the digits 1 and 2.

You can make 12 (Niven) and 21 (Niven).
Now, add a zero. You can make 102, 120, 201, 210, 012 (which is just 12), and 021 (which is just 21).
The authors found that if you have a specific set of digits (like 1, 2, and a bunch of zeros), you can shuffle them around, and every resulting number will still be a Niven number. It's like having a deck of cards where every hand you deal is a winning hand.

3. The "Repeating" Numbers
They also looked at numbers made of repeating digits, like 111, 222, or 5555.

If 111 is a Niven number (it is: $1+1+1=3$ , and $111 \div 3 = 37$ ), then it's automatically a PINN. Why? Because if you shuffle the digits of 111, you just get 111 again! It never changes.
They found a mathematical formula to generate infinite families of these repeating numbers that are always PINNs.

4. The "Shuffle" Algorithm
The authors created a step-by-step recipe (an algorithm) to find these numbers.

Step 1: Find numbers made of non-zero digits that work.
Step 2: Take those working numbers and insert zeros in different spots.
Step 3: Check if the new numbers still work.
They used this method to list out all the PINNs with up to 9 digits. It's like a computer program that sorts through millions of combinations to find the "golden tickets."

Why Does This Matter?

You might ask, "Who cares about shuffling digits?"

In the world of math, this is like studying symmetry.

In physics, scientists study how particles behave when you rotate them or flip them. If a particle looks the same after a flip, it has "symmetry."
In this paper, the "particles" are digits, and the "flip" is shuffling them.
Finding numbers that stay "nice" (Niven) no matter how you shuffle them helps mathematicians understand the deep, hidden rules of how numbers are built. It's like discovering that certain Lego structures are so perfectly balanced that you can take them apart and rebuild them in any shape, and they will still hold together.

The Big Picture

The paper concludes with a few open questions, like:

Does this work in other number systems (like binary, which computers use)?
Are there any PINNs that are also prime numbers (numbers divisible only by 1 and themselves)? (The authors found some very large, rare prime PINNs!).

In a nutshell:
This paper introduces a new class of "indestructible" numbers. Just as a well-designed puzzle piece fits into a box no matter how you turn it, these numbers remain "divisible by their digit sum" no matter how you rearrange their digits. The authors proved there are infinite of them, showed you how to find them, and gave us a glimpse into the beautiful, symmetrical patterns hidden inside our number system.

1. Problem Statement

The paper introduces and investigates a novel class of integers called Permutation-Invariant Niven Numbers (PINNs).

Context: A standard Niven number (or Harshad number) is an integer divisible by the sum of its digits. While the properties of standard Niven numbers are well-studied (e.g., their density is zero, and arbitrarily long sequences exist), their behavior under digit permutation has not been systematically analyzed.
Core Question: The authors ask: What are the properties of integers such that every permutation of their digits (ignoring leading zeros) results in a Niven number?
Specific Research Questions: The paper addresses ten specific questions, including:
- What is the smallest PINN?
- Are all repdigits (e.g., 111, 222) PINNs?
- Can PINNs contain the digit 0?
- Do infinitely many PINNs exist, and what is their asymptotic density?
- What are the necessary and sufficient conditions for a number to be a PINN?
- How can PINNs be algorithmically generated?

2. Methodology

The authors employ a combination of combinatorial number theory, modular arithmetic, and exhaustive computational search.

Definitions and Group Theory:
- PINNs are defined formally using the symmetric group $S_k$ . A $k$ -digit number is a PINN if every permutation $\sigma \in S_k$ applied to its digits yields a number divisible by the digit sum $s$ .
- The authors adopt the convention that leading zeros are ignored (e.g., $012$ is treated as $12$), allowing PINNs to contain internal or trailing zeros.
Two-Stage Search Algorithm:
To generate PINNs for a specific digit length $k$ $k$ , the authors propose a two-step algorithm:
1. Non-zero Search: Identify all PINNs composed entirely of $k$ non-zero digits.
2. Zero Augmentation: Systematically insert zeros into smaller PINNs (of length $j < k$ ) to form $k$ -digit numbers, verifying if the Niven property is preserved.
Congruence Analysis:
The paper utilizes modular arithmetic to prove that specific families of numbers satisfy the divisibility condition for all permutations. This involves analyzing the digit sum $s$ and the number modulo $s$ .
Computational Verification:
The authors performed exhaustive searches for small $k$ (up to $k=9$ ) and verified a conjecture regarding repdigit PINNs up to a magnitude of $\sim 10^{108}$ .

3. Key Contributions and Results

A. Existence and Density

Infinitude: The paper proves that there are infinitely many PINNs. This is demonstrated by constructing infinite families of numbers (specifically repdigits and numbers with specific zero-insertion patterns).
Density: The asymptotic (natural) density of PINNs is zero. Since PINNs are a strict subset of Niven numbers (which already have a density of zero), the density of PINNs must also be zero.

B. Classification of Small PINNs

The authors provide exhaustive lists of PINNs for small digit lengths:

1-digit: All digits 1–9 are PINNs.
2-digits: 23 specific numbers (e.g., 10, 12, 18, 20, ...).
3-digits: Categorized into four sets ( $N_{31}$ to $N_{34}$ ), including repdigits (111, 222...), numbers with zeros (100, 120...), and mixed digits (117, 126...).
4–9 digits: The authors provide a structured classification ( $N_{k1}$ to $N_{kj}$ ) showing that for $k \ge 4$ , PINNs are largely formed by appending zeros to smaller PINNs or using specific non-zero digit combinations.

C. Repdigit PINNs and Conjectures

Repdigits: Not all repdigits are Niven numbers, but those that are (e.g., $111$ where $1+1+1=3$ ) are trivially PINNs because all permutations are identical.
Infinite Families: The authors propose Conjecture 1, providing a parametric solution for the congruence condition required for repdigit PINNs. The conjecture suggests an infinite family of solutions parameterized by indices involving specific large primes (e.g., $m_0=37, m_1=333667$ ).
Zero Insertion: The paper demonstrates that while zeros can be inserted into some PINNs (e.g., $12 \to 102$ ), they cannot be arbitrarily inserted into all PINNs. Specifically, inserting zeros into certain repdigit PINNs (like $1^{27}$ ) breaks the Niven property.

D. Main Theorem (Theorem 1)

The central result is Theorem 1, which characterizes all $k$ -digit PINNs for $k \ge 10$ .

The theorem states that all $k$ -digit PINNs belong to the union of 10 specific sets ( $N_{ka}$ through $N_{kj}$ ) under the action of the symmetric group $S_k$ .
These sets are constructed by taking specific "seed" numbers (e.g., $10, 120, 117, 1116$) and appending a specific number of zeros ( $k-i$ zeros) to them.
The proof relies on showing that for any permutation of these constructed numbers, the resulting value is divisible by the digit sum (which is constant for a given set).

E. Primes and OEIS Relations

The authors identify a subset of primes related to the parameters of the repdigit PINN conjecture (e.g., 37, 163, 757, 333667).
They note that while PINNs are a subset of Niven numbers, no direct mapping to existing OEIS sequences (other than Niven numbers themselves) has been found yet.
No PINNs that are also prime numbers (other than single digits) or palindromes (other than repdigits) have been discovered in the exhaustive search.

4. Significance

New Mathematical Object: The paper defines a new class of integers with strong symmetry properties, bridging combinatorics (permutations) and number theory (divisibility).
Structural Insight: It reveals that the set of PINNs is highly structured. Unlike general Niven numbers, which appear somewhat randomly, PINNs for $k \ge 10$ follow a rigid pattern of "seed numbers + zeros."
Algorithmic Efficiency: The proposed two-stage search algorithm offers a more efficient method for generating these numbers than brute-force checking, leveraging the invariance property to reduce the search space.
Foundation for Future Work: The paper opens several avenues for research, including the extension of PINNs to other bases (binary, ternary), the rigorous proof of Conjecture 1, and the search for PINNs that are also prime or palindromic.

5. Conclusion

Wu and Lou successfully establish the existence of infinitely many Permutation-Invariant Niven Numbers. They provide a complete classification for numbers up to 9 digits and a general theorem for $k \ge 10$ , showing that these numbers are formed by specific digit patterns augmented with zeros. The work highlights that while the density of these numbers is zero, their structural properties are rich and amenable to algebraic characterization.