Permutation-Invariant Niven Numbers

Abstract

This paper introduces permutation-invariant Niven numbers (PINNs), a novel class of Niven numbers where all digit permutations (with leading zeros automatically ignored) must retain the Niven property. We demonstrate that there exist infinitely many such numbers and that their magnitude is unbounded. Furthermore, we present an exhaustive search method for identifying permutation-invariant Niven numbers. Complete classifications for digit lengths up to 9 are provided, and an infinite family for arbitrary digit lengths is constructed. The asymptotic density of PINNs is shown to be zero, and various arithmetic and combinatorial properties are investigated. We present a novel parameterization of infinitely many repdigit PINNs through Conjecture 1, which establishes an explicit multiplicative structure involving distinguished prime factors derived from repunit arithmetic. This provides a systematic method for generating infinitely many new repdigit Niven numbers.

1. Introduction

Niven numbers (also known as Harshad numbers)—integers divisible by the sum of their digits—were first conceptualized by Niven in a 1977 lecture [1], where he explored their existence properties within base-10. The term “Niven number (NN)” was later coined by Kennedy [2], who formalized their study by establishing their natural density (zero) and constructing infinitely many such numbers. These seminal works catalyzed research into the combinatorial and arithmetic properties of digit-sum-dependent sequences, extending to generalizations across bases, gaps, additive representations, and asymptotic distributions.

Since the pioneering works of Niven and Kennedy [1,2], various studies have been conducted by many authors. Cooper and Kennedy prove the existence of arbitrarily long sequences of consecutive Niven numbers [3]. Grundman refines Cooper–Kennedy’s theorem by establishing explicit bounds for the minimal starting point of k-consecutive Niven sequences [4]. De Koninck and Doyon provide a rigorous asymptotic formula $N\left(x\right)\sim c{\displaystyle \frac{x}{logx}}$ for the counting function of NNs [5]. Cai characterizes Niven numbers in bases 2 and 3 [6], revealing structural properties unique to low bases. Fredricksen, Ionascu and Luca solve the minimality problem for NNs with fixed digit sum s [7]. Sanna presents a novel approach combining deep tools from analytic number theory with combinatorial digit sum manipulations to establish the additive basis property for Niven numbers [8]. Wilson provides a valuable constructive proof explicitly demonstrating the existence of exceptionally small sequences of 20 consecutive NNs [9].

Despite extensive research on Niven numbers, the permutation invariance property—requiring all digit permutations of a number to retain the Niven property—has not been systematically investigated. This paper introduces and studies permutation-invariant Niven numbers (PINNs), a structured subclass with strong combinatorial constraints. Our work makes four key contributions:

• We prove the existence of infinitely many PINNs and establish their zero asymptotic density.
• We develop an efficient two-step search algorithm that avoids factorial blowup by exploiting digit-sum congruence and symmetry, providing complete classifications for digit lengths up to 9.
• We construct an infinite family of PINNs for arbitrary digit lengths and derive necessary digit-sum conditions.
• We present a novel parameterization of infinitely many repdigit PINNs through Conjecture 1, which reveals a multiplicative structure involving special primes that are primitive divisors of repunits, establishing a novel connection between permutation invariance in Niven numbers and the arithmetic of recurring decimals. This parameterization not only generates infinitely many new Niven numbers but also provides a systematic method for constructing them through distinguished prime factors.

The paper is organized as follows: Section 2 provides formal definitions and notation. Section 3 lists the research questions guiding our investigation. Section 4 examines basic properties of PINNs, including the innovative parameterization of repdigit PINNs. Section 5 presents exhaustive search results for small digit lengths. Section 6 details the algorithmic generation method. Section 7 presents the main theorem constructing infinite families. Section 8 discusses further properties and conditions. Section 9 concludes with open problems.

2. Definitions and Preliminaries

2.1. Notation Summary

For clarity, we summarize key notations in

Table 1. Summary of notations used throughout the paper.

Symbol	Meaning
$A_k=a_k{a}_{k-1}\dots a_1$	k-digit integer with digits $a_i\in \{0,\dots ,9\}$, $a_k\ne 0$
${\mathcal{S}}_k$	Symmetric group on k elements
${\tau }_i=(1,i)$	Transposition swapping positions 1 and i
${\mathcal{A}}_k$	Set of all digit permutations of $A_k$ (leading zeros removed)
${\mathrm{NN}}_k$	Set of k-digit Niven numbers
${\mathrm{PINN}}_k$	Set of k-digit permutation-invariant Niven numbers
$a_{\left(n\right)}$	n-digit repdigit consisting of digit a (e.g., $1_{\left(3\right)}=111$)
$d_{\mathrm{PINN}}$	Asymptotic (natural) density of PINNs
$0_{\left(m\right)}$	m consecutive zeros (used in concatenation)

.

2.2. Formal Definition of PINNs

Definition 1.

A permutation-invariant Niven number (PINN) is a positive integer N such that for every permutation π of its decimal digits (after removing any leading zeros), the resulting integer $\pi \left(N\right)$ is a Niven number.

For a k-digit number $A_k=a_k{a}_{k-1}\dots a_1$ (where $a_k$ is the most significant digit), the set of all digit permutations (with leading zeros removed) is denoted by

$${\mathcal{A}}_k=\left\{\pi \left(A_k\right)\mid \pi \in {\mathcal{S}}_k\right\},$$

where ${\mathcal{S}}_k$ denotes the symmetric group on k elements. This group is generated by transpositions

$${\mathcal{S}}_k=\langle {\tau }_2,{\tau }_3,\dots ,{\tau }_k\rangle ,{\tau }_i=(1,i)\mathrm{for}i=2,\dots ,k,$$

where each generator ${\tau }_i$ acts on $A_k$ by swapping the digits at positions 1 and i:

$${\tau }_i(a_k\dots a_i\dots a_2{a}_1)=a_k\dots a_1\dots a_2{a}_i.$$

We adopt the convention that leading zeros are automatically ignored, i.e.,

$$0_{(i-1)}{a}_i\dots a_k=a_i\dots a_k\mathrm{for}{a}_i\ne 0,$$

where $0_{(i-1)}$ denotes $i-1$ leading zeros.

3. Research Questions

The following questions guide our investigation of permutation-invariant Niven numbers:

(1)

What is the smallest PINN?

(2)

Are all repdigits (e.g., 111, 222, …) PINNs?

(3)

Can a number containing the digit 0 be PINN?

(4)

Do infinitely many PINNs exist? If yes, what is their asymptotic density?

(5)

For fixed digit-length k, how many PINNs exist?

(6)

How can we algorithmically generate all PINNs?

(7)

What are necessary and sufficient conditions for such numbers?

(8)

What conditions must the digit sum s satisfy to ensure that all permutations (without leading zeros) of a digit set are divisible by s?

(9)

How do these numbers relate to established OEIS sequences (e.g., NNs, repdigits)? Do subclasses (primes, palindromes) exist?

(10)

What open questions remain? Can this concept extend to other bases (e.g., binary)?

Answers to these questions are provided throughout the paper, with explicit references given in Section 9.

4. Basic Properties

4.1. Single-Digit PINNs

For the case of single-digit numbers, it is straightforward that all are PINNs. Specifically,

$${\mathrm{NN}}_1\equiv \{1,2,3,4,5,6,7,8,9\},$$

where ${\mathrm{NN}}_1$ denotes the set of 1-digit PINNs. We note that 1 is the smallest element in this set, answering question (1).

4.2. Repdigits as PINNs: A Novel Parameterization

Regarding question (2), we first observe that not all repdigits are Niven numbers, and consequently, not all repdigits are PINNs. However, if a repdigit is an NN, it trivially qualifies as a PINN since its digit permutations are identical. For example, any 3-digit repdigit $aaa$ with $a\in {\mathrm{NN}}_1$ is an NN because

$$aaa\equiv 0\left(\mathrm{mod}3a\right).$$

For notational convenience in subsequent analysis, we define $a_{\left(n\right)}$ as the n-digit repdigit formed by repeating digit a. Representative cases include

$$1_{\left(3\right)}=111,5_{\left(4\right)}=5555.$$

It is established that infinitely many repdigit NNs (which are simultaneously PINNs) exist. These admit the general form

$$N={\displaystyle \frac{({10}^k-1)a}{9}},a\in {\mathrm{NN}}_1,{10}^k\equiv 1\left(\mathrm{mod}9k\right).$$

(1)

To explicitly demonstrate the existence of infinitely many such repdigit NNs/PINNs, we propose the following solution conjecture for the congruence condition:

Conjecture 1 (Parameterization of Repdigit PINNs).

The repdigit NN Equation (1) admits solutions parameterized as

$$\begin{array}{c}k=k_{\left\{n\alpha \beta \gamma \delta \right\}}=3^n·m_0^{\alpha }·m_1^{\beta }·m_{21}^{{\gamma }_1}·m_{22}^{{\gamma }_2}·m_{31}^{{\delta }_1}·m_{32}^{{\delta }_2}·m_{33}^{{\delta }_3}·m_{34}^{{\delta }_4}·m_{35}^{{\delta }_5}\hfill \\ for\left\{\alpha \beta \gamma \delta \right\}\equiv \{\alpha ,\beta ,{\gamma }_1,{\gamma }_2,{\delta }_1,{\delta }_2,\dots ,{\delta }_5\}\in {\mathbb{N}}_0=\{0,1,2,\dots \},\hfill \end{array}$$

subject to the following minimal-index constraints,

$$\begin{array}{cc}\hfill & n\ge 0when\left\{\alpha \beta \gamma \delta \right\}=\left\{0\right\},\hfill \\ \hfill & n\ge 1when\alpha \ne 0,\hfill \\ \hfill & n\ge 2when\beta \ne 0,\hfill \\ \hfill & n\ge 3when\gamma =\{{\gamma }_1,{\gamma }_2\}\ne \left\{0\right\},\hfill \\ \hfill & n\ge 4when\delta =\{{\delta }_1,\dots ,{\delta }_5\}\ne \left\{0\right\},\hfill \end{array}$$

where $m_0=37$, $m_1=333667$, $m_{21}=757$, $m_{22}=440334654777631$, $m_{31}=163$, $m_{32}=9937$, $m_{33}=2462401$, $m_{34}=676421558270641$, and $m_{35}=130654897808007778425046117$.

Significance of Conjecture 1

Conjecture  1 represents a significant innovation in the study of Niven numbers for several reasons:

1.

Systematic Generation: It provides an explicit, multiplicative parameterization of infinitely many repdigit PINNs, going beyond mere existence proofs.

2.

Connection to Repunit Primes: The parameters $m_i$ are not arbitrary primes but are precisely primitive prime divisors of repunits $R_n=({10}^n-1)/9$. This establishes a deep connection between permutation invariance in Niven numbers and the arithmetic of recurring decimals.

3.

Infinite Family: By freely varying the exponents $n,\alpha ,\beta ,{\gamma }_i,{\delta }_i$, we obtain infinitely many distinct exponents k, and consequently infinitely many distinct repdigit PINNs for each digit a.

4.

Computational Verification: The conjecture has been verified computationally up to our system’s overflow threshold of ∼${10}^{{10}^8}$, providing strong empirical support.

5.

Partial Rigorous Proof: A special case of the conjecture (with ${\gamma }_2={\delta }_4={\delta }_5=0$) can be rigorously proved, indicating that the full conjecture is plausible and lies within reach of known number-theoretic techniques.

6.

New Niven Numbers: Even for the previously studied class of repdigit Niven numbers, our parameterization yields infinitely many new examples that were not previously identified, as the exponents k go far beyond those found by brute-force search.

This parameterization reveals that the divisibility condition ${10}^k\equiv 1\left(\mathrm{mod}9k\right)$ is intimately tied to the multiplicative order of 10 modulo these special primes. The appearance of these primes in the factorization of k guarantees that the resulting repdigit satisfies both the Niven and permutation invariance properties.

Under this conjecture, the repdigit numbers $a_{\left(k_{n\alpha \beta \gamma \delta }\right)}$ with $a\in {\mathrm{NN}}_1$ are both NNs and PINNs since they satisfy

$$a_{\left(k_{n\alpha \beta \gamma \delta }\right)}={\displaystyle \frac{a}{9}}({10}^{k_{n\alpha \beta \gamma \delta }}-1)\equiv 0(\mathrm{mod}a·k_{n\alpha \beta \gamma \delta }).$$

4.3. Numbers Containing Zero

For question (3), PINNs may contain internal or trailing zeros under our convention that leading zeros are omitted.

As an example, consider numbers of the form ${10}_{\left(n_1\right)}{20}_{\left(n_2\right)}$ and ${20}_{\left(m_1\right)}{10}_{\left(m_2\right)}$ for nonnegative integers $n_1,n_2,m_1,m_2$. Since any permutation of their digits (after removing leading zeros) has digit sum $1+2=3$, and because

$${10}_{\left(n_1\right)}{20}_{\left(n_2\right)}\equiv 0,{20}_{\left(m_1\right)}{10}_{\left(m_2\right)}\equiv 0\left(\mathrm{mod}3\right),$$

where (by the congruence property of digit sums modulo 3) all permutations yield Niven numbers. Thus, these numbers are PINNs.

4.4. Infinitude and Density

For question (4), the answer is affirmative because the arbitrariness of the integers in the constructions above demonstrates the existence of infinitely many PINNs.

The density of PINNs is defined as their asymptotic density (or natural density), given by the limit of the proportion of PINNs among the first N positive integers as $N\to \infty$. Denoting $N_{\mathrm{PINNs}}\left(N\right)$ and $N_{\mathrm{NNs}}\left(N\right)$ as the counts of PINNs and NNs up to N respectively, we have

$$d_{\mathrm{PINNs}}=\underset{N\to \infty }{lim}{\displaystyle \frac{N_{\mathrm{PINNs}}\left(N\right)}{N}}\le \underset{N\to \infty }{lim}{\displaystyle \frac{N_{\mathrm{NNs}}\left(N\right)}{N}}=d_{\mathrm{NNs}}=0,$$

(2)

implying that the density of PINNs is zero. The inequality follows because PINNs constitute a subset of NNs, while the final equality is a known result for NNs [2].

5. Exhaustive Search for Small PINNs

5.1. 2-Digit PINNs

For $k=2$, verification confirms that the complete set of PINNs consists of

$${\mathrm{NN}}_2\equiv \{10,12,18,20,21,24,27,30,36,40,42,45,48,50,54,60,63,70,72,80,81,84,90\}.$$

Using the convention that leading zeros are ignored, the union of ${\mathrm{NN}}_1$ and ${\mathrm{NN}}_2$ can be defined as

$$\begin{array}{cc}\hfill {\mathrm{PINN}}_2& \equiv {\mathrm{NN}}_1\cup {\mathrm{NN}}_2={\mathcal{S}}_2{\mathcal{N}}_2,\hfill \\ \hfill {\mathcal{N}}_2& =\{10,12,18,20,24,27,30,36,40,45,48,50,60,70,80,90\}.\hfill \end{array}$$

The ${\mathcal{S}}_2$-invariance of the 2-digit PINNs defined in ${\mathcal{N}}_2$ follows directly from their construction, as expressed by the symmetry relation:

$${\mathcal{S}}_2\left({\mathrm{PINN}}_2\right)={\mathrm{PINN}}_2.$$

Furthermore, the inclusion hierarchy

$${\mathrm{PINN}}_2\supset {\mathrm{PINN}}_1={\mathrm{NN}}_1$$

holds as a consequence of the notational convention.

5.2. 3-Digit PINNs

After detailed verification, we establish that the 3-digit PINNs can be expressed as

$$\begin{array}{cc}\hfill {\mathrm{PINN}}_3& \equiv {\mathcal{S}}_3{\mathcal{N}}_3={\mathcal{S}}_3\left({\mathcal{N}}_{31}\cup {\mathcal{N}}_{32}\cup {\mathcal{N}}_{33}\cup {\mathcal{N}}_{34}\right),\hfill \\ \hfill {\mathcal{N}}_{31}& =\{100,200,300,400,500,600,700,800,900\},\hfill \\ \hfill {\mathcal{N}}_{32}& =\{120,180,240,270,360,450,480\},\hfill \\ \hfill {\mathcal{N}}_{33}& =\{1_{\left(3\right)},2_{\left(3\right)},3_{\left(3\right)},4_{\left(3\right)},5_{\left(3\right)},6_{\left(3\right)},7_{\left(3\right)},8_{\left(3\right)},9_{\left(3\right)}\},\hfill \\ \hfill {\mathcal{N}}_{34}& =\{117,126,135,144,225,234,288,468\}.\hfill \end{array}$$

The ${\mathcal{S}}_3$-invariance of ${\mathrm{PINN}}_3$ is manifest:

$${\mathcal{S}}_3{\mathrm{PINN}}_3={\mathrm{PINN}}_3.$$

Furthermore, when accounting for the convention that leading zeros are ignored, we immediately deduce the containment relation:

$${\mathrm{PINN}}_3\supset {\mathrm{PINN}}_2.$$

6. Algorithmic Generation of PINNs

We now address question (6) by introducing a general algorithm for generating all PINNs. To find k-digit PINNs, we exploit their permutation invariance property through a two-stage search algorithm (Algorithm  1):

6.1. Complexity Analysis and Implementation Details

The naive approach of checking all $k!$ permutations for each candidate is infeasible for large k. Our algorithm reduces complexity through several optimizations:

• Multiset enumeration: Instead of checking all ${10}^k$ numbers, we enumerate digit multisets. For k digits chosen from $\{0,\dots ,9\}$ with at most k copies of each digit, the number of multisets is $\left({\displaystyle \genfrac{}{}{0pt}{}{k+10-1}{10-1}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{k+9}{9}}\right)$, which grows polynomially in k.
• Representative testing: For a given digit multiset with sum s, we only need to test one representative permutation (e.g., the largest number) modulo s. However, careful analysis shows that for permutation invariance, we must verify that all permutations are divisible by s. We can prune the search by noting that if s has factors 2 or 5, the last digit matters, etc.
• Zero-insertion reuse: The second stage reuses previously validated PINNs of shorter lengths, avoiding redundant checks.

For $k\le 9$, exhaustive search is feasible with efficient implementation. For larger k, only structural families (as in Theorem 1) are known.

Algorithm 1 Two-stage search for k-digit PINNs

Require: Digit length $k\ge 1$

Ensure: Set ${\mathrm{PINN}}_k$ of all k-digit PINNs

1: $\mathcal{C}\leftarrow \varnothing$                                                                                                                                 ▹ Initialize result set

2:

3: procedure Stage1(k)                                                                   ▹ Search for PINNs with all digits nonzero

4:       for each multiset M of k nonzero digits do

5:             Compute digit sum $s={\sum }_{d\in M}d$

6:             Let $N_0$ be the integer formed by sorting digits descending

7:             if $N_0\equiv 0\left(\mathrm{mod}s\right)$ then

8:                  Check if all permutations of M yield Niven numbers

9:                  if condition holds then

10:                      Add all permutations to $\mathcal{C}$

11:                end if

12:             end if

13:         end for

14: end procedure

15:

16: procedure Stage2(k)                                                                           ▹ Augment shorter PINNs with zeros

17:         for each j-digit PINN $P\in {\mathrm{PINN}}_j$ with $1\le j<k$ do

18:           for each placement of $k-j$ zeros (no leading zero) do

19:                Let N be the resulting k-digit number

20:                if all digit permutations of N are Niven numbers then

21:                    Add all permutations to $\mathcal{C}$

22:               end if

23:            end for

24:       end for

25: end procedure

26:

27: Call Stage1$\left(k\right)$

28: Call Stage2$\left(k\right)$

29: return  $\mathcal{C}$

6.2. Application to $k=4$

For $k=4$, one can verify that the PINNs containing no zero digits consist of precisely the following 12 elements:

$${\mathcal{N}}_{45}=\{1116,1125,1134,1224,1233,2223,2448,2268,2466,3699,4446,6669\}.$$

The next step demonstrates that appending one or more zeros to any element of ${\mathrm{PINN}}_3$ preserves the Niven property. Thus, the ${\mathcal{S}}_4$-invariant PINNs take the form:

$${\mathrm{PINN}}_4={\mathcal{S}}_4{\mathcal{N}}_4={\mathcal{S}}_4\left({\mathcal{N}}_{41}\cup {\mathcal{N}}_{42}\cup {\mathcal{N}}_{43}\cup {\mathcal{N}}_{44}\cup {\mathcal{N}}_{45}\right),$$

where ${\mathcal{N}}_{45}$ is given above and

$$\begin{array}{cc}\hfill {\mathcal{N}}_{41}& =\{{10}_{\left(3\right)},{20}_{\left(3\right)},{30}_{\left(3\right)},{40}_{\left(3\right)},{50}_{\left(3\right)},{60}_{\left(3\right)},{70}_{\left(3\right)},{80}_{\left(3\right)},900\},\hfill \\ \hfill {\mathcal{N}}_{42}& =\{1200,1800,2400,2700,3600,4500,4800\},\hfill \\ \hfill {\mathcal{N}}_{43}& =\{1_{\left(3\right)}0,2_{\left(3\right)}0,3_{\left(3\right)}0,4_{\left(3\right)}0,5_{\left(3\right)}0,6_{\left(3\right)}0,7_{\left(3\right)}0,8_{\left(3\right)}0,9_{\left(3\right)}0\},\hfill \\ \hfill {\mathcal{N}}_{44}& =\{1170,1260,1350,1440,2250,2340,2880,4680\}.\hfill \end{array}$$

6.3. Higher-Digit PINNs ($k=5$ to 9)

Applying the two-step search algorithm to k-digit PINNs for $k=5,6,\dots ,9$, we obtain:

• 5-digit PINNs:

$$\begin{array}{cc}\hfill {\mathrm{PINN}}_5& \equiv {\mathcal{S}}_5{\mathcal{N}}_5={\mathcal{S}}_5({\mathcal{N}}_{51}\cup {\mathcal{N}}_{52}\cup {\mathcal{N}}_{53}\cup {\mathcal{N}}_{54}\cup {\mathcal{N}}_{55}\cup {\mathcal{N}}_{56}),\hfill \\ \hfill {\mathcal{N}}_{51}& =\{{10}_{\left(4\right)},{20}_{\left(4\right)},{30}_{\left(4\right)},{40}_{\left(4\right)},{50}_{\left(4\right)},{60}_{\left(4\right)},{70}_{\left(4\right)},{80}_{\left(4\right)},{90}_{\left(4\right)}\},\hfill \\ \hfill {\mathcal{N}}_{52}& =\{{120}_{\left(3\right)},{180}_{\left(3\right)},{240}_{\left(3\right)},{270}_{\left(3\right)},{360}_{\left(3\right)},{450}_{\left(3\right)},{480}_{\left(3\right)}\},\hfill \\ \hfill {\mathcal{N}}_{53}& =\{1_{\left(3\right)}00,2_{\left(3\right)}00,3_{\left(3\right)}00,4_{\left(3\right)}00,5_{\left(3\right)}00,6_{\left(3\right)}00,7_{\left(3\right)}00,8_{\left(3\right)}00,9_{\left(3\right)}00\},\hfill \\ \hfill {\mathcal{N}}_{54}& =\{11700,12600,13500,14400,22500,23400,28800,46800\},\hfill \\ \hfill {\mathcal{N}}_{55}& =\{1_{\left(3\right)}60,11250,11340,12240,12330,2_{\left(3\right)}30,24480,22680,24660,36990,4_{\left(3\right)}60,6_{\left(3\right)}90\},\hfill \\ \hfill {\mathcal{N}}_{56}& =\{1_{\left(4\right)}5,1_{\left(3\right)}24,1_{\left(3\right)}33,11223,{12}_{\left(4\right)},2_{\left(3\right)}48,2_{\left(3\right)}66,22446,{24}_{\left(4\right)},3_{\left(3\right)}99,33669,{36}_{\left(4\right)},{48}_{\left(4\right)}\}.\hfill \end{array}$$

• 6-digit PINNs:

$$\begin{array}{cc}\hfill {\mathrm{PINN}}_6& \equiv {\mathcal{S}}_6{\mathcal{N}}_6={\mathcal{S}}_6({\mathcal{N}}_{61}\cup {\mathcal{N}}_{62}\cup {\mathcal{N}}_{63}\cup {\mathcal{N}}_{64}\cup {\mathcal{N}}_{65}\cup {\mathcal{N}}_{66}\cup {\mathcal{N}}_{67}),\hfill \\ \hfill {\mathcal{N}}_{61}& =\{{10}_{\left(5\right)},{20}_{\left(5\right)},{30}_{\left(5\right)},{40}_{\left(5\right)},{50}_{\left(5\right)},{60}_{\left(5\right)},{70}_{\left(5\right)},{80}_{\left(5\right)},{90}_{\left(5\right)}\},\hfill \\ \hfill {\mathcal{N}}_{62}& =\{{120}_{\left(4\right)},{180}_{\left(4\right)},{240}_{\left(4\right)},{270}_{\left(4\right)},{360}_{\left(4\right)},{450}_{\left(4\right)},{480}_{\left(4\right)}\},\hfill \\ \hfill {\mathcal{N}}_{63}& =\{1_{\left(3\right)}{0}_{\left(3\right)},2_{\left(3\right)}{0}_{\left(3\right)},3_{\left(3\right)}{0}_{\left(3\right)},4_{\left(3\right)}{0}_{\left(3\right)},5_{\left(3\right)}{0}_{\left(3\right)},6_{\left(3\right)}{0}_{\left(3\right)},7_{\left(3\right)}{0}_{\left(3\right)},8_{\left(3\right)}{0}_{\left(3\right)},9_{\left(3\right)}{0}_{\left(3\right)}\},\hfill \\ \hfill {\mathcal{N}}_{64}& =\{{1170}_{\left(3\right)},{1260}_{\left(3\right)},{1350}_{\left(3\right)},{1440}_{\left(3\right)},{2250}_{\left(3\right)},{2340}_{\left(3\right)},{2880}_{\left(3\right)},{4680}_{\left(3\right)}\},\hfill \\ \hfill {\mathcal{N}}_{65}& =\{1_{\left(3\right)}{60}_{\left(2\right)},{11250}_{\left(2\right)},{11340}_{\left(2\right)},{12240}_{\left(2\right)},{12330}_{\left(2\right)},2_{\left(3\right)}{30}_{\left(2\right)},\hfill \\ \hfill & {24480}_{\left(2\right)},{22680}_{\left(2\right)},{24660}_{\left(2\right)},{36990}_{\left(2\right)},4_{\left(3\right)}{60}_{\left(2\right)},6_{\left(3\right)}{90}_{\left(2\right)}\},\hfill \\ \hfill {\mathcal{N}}_{66}& =\{1_{\left(4\right)}{50}_{\left(1\right)},1_{\left(3\right)}{240}_{\left(1\right)},1_{\left(3\right)}{330}_{\left(1\right)},{112230}_{\left(1\right)},{12}_{\left(4\right)}{0}_{\left(1\right)},2_{\left(3\right)}{480}_{\left(1\right)},\hfill \\ \hfill & 2_{\left(3\right)}{660}_{\left(1\right)},{224460}_{\left(1\right)},{24}_{\left(4\right)}{0}_{\left(1\right)},3_{\left(3\right)}{990}_{\left(1\right)},{336690}_{\left(1\right)},{36}_{\left(4\right)}{0}_{\left(1\right)},{48}_{\left(4\right)}{0}_{\left(1\right)}\},\hfill \\ \hfill {\mathcal{N}}_{67}& =\{1_{\left(5\right)}4,1_{\left(4\right)}23,1_{\left(3\right)}{2}_{\left(3\right)},2_{\left(5\right)}8,2_{\left(4\right)}46,2_{\left(3\right)}{4}_{\left(3\right)},3_{\left(4\right)}69,3_{\left(3\right)}{6}_{\left(3\right)},4_{\left(3\right)}{8}_{\left(3\right)}\}.\hfill \end{array}$$

• 7-digit PINNs:

$$\begin{array}{cc}\hfill {\mathrm{PINN}}_7& \equiv {\mathcal{S}}_7{\mathcal{N}}_7={\mathcal{S}}_7({\mathcal{N}}_{71}\cup {\mathcal{N}}_{72}\cup {\mathcal{N}}_{73}\cup {\mathcal{N}}_{74}\cup {\mathcal{N}}_{75}\cup {\mathcal{N}}_{76}\cup {\mathcal{N}}_{77}\cup {\mathcal{N}}_{78}),\hfill \\ \hfill {\mathcal{N}}_{71}& =\{{10}_{\left(6\right)},{20}_{\left(6\right)},{30}_{\left(6\right)},{40}_{\left(6\right)},{50}_{\left(6\right)},{60}_{\left(6\right)},{70}_{\left(6\right)},{80}_{\left(6\right)},{90}_{\left(6\right)}\},\hfill \\ \hfill {\mathcal{N}}_{72}& =\{{120}_{\left(5\right)},{180}_{\left(5\right)},{240}_{\left(5\right)},{270}_{\left(5\right)},{360}_{\left(5\right)},{450}_{\left(5\right)},{480}_{\left(5\right)}\},\hfill \\ \hfill {\mathcal{N}}_{73}& =\{1_{\left(3\right)}{0}_{\left(4\right)},2_{\left(3\right)}{0}_{\left(4\right)},3_{\left(3\right)}{0}_{\left(4\right)},4_{\left(3\right)}{0}_{\left(4\right)},5_{\left(3\right)}{0}_{\left(4\right)},6_{\left(3\right)}{0}_{\left(4\right)},7_{\left(3\right)}{0}_{\left(4\right)},8_{\left(3\right)}{0}_{\left(4\right)},9_{\left(3\right)}{0}_{\left(4\right)}\},\hfill \\ \hfill {\mathcal{N}}_{74}& =\{{1170}_{\left(4\right)},{1260}_{\left(4\right)},{1350}_{\left(4\right)},{1440}_{\left(4\right)},{2250}_{\left(4\right)},{2340}_{\left(4\right)},{2880}_{\left(4\right)},{4680}_{\left(4\right)}\},\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathcal{N}}_{75}& =\{1_{\left(3\right)}{60}_{\left(3\right)},{11250}_{\left(3\right)},{11340}_{\left(3\right)},{12240}_{\left(3\right)},{12330}_{\left(3\right)},2_{\left(3\right)}{30}_{\left(3\right)},\hfill \\ \hfill & {24480}_{\left(3\right)},{22680}_{\left(3\right)},{24660}_{\left(3\right)},{36990}_{\left(3\right)},4_{\left(3\right)}{60}_{\left(3\right)},6_{\left(3\right)}{90}_{\left(3\right)}\},\hfill \\ \hfill {\mathcal{N}}_{76}& =\{1_{\left(4\right)}{50}_{\left(2\right)},1_{\left(3\right)}{240}_{\left(2\right)},1_{\left(3\right)}{330}_{\left(2\right)},{112230}_{\left(2\right)},{12}_{\left(4\right)}{0}_{\left(2\right)},2_{\left(3\right)}{480}_{\left(2\right)},\hfill \\ \hfill & 2_{\left(3\right)}{660}_{\left(2\right)},{224460}_{\left(2\right)},{24}_{\left(4\right)}{0}_{\left(2\right)},3_{\left(3\right)}{990}_{\left(2\right)},{336690}_{\left(2\right)},{36}_{\left(4\right)}{0}_{\left(2\right)},{48}_{\left(4\right)}{0}_{\left(2\right)}\},\hfill \\ \hfill {\mathcal{N}}_{77}& =\{1_{\left(5\right)}{40}_{\left(1\right)},1_{\left(4\right)}{230}_{\left(1\right)},1_{\left(3\right)}{2}_{\left(3\right)}{0}_{\left(1\right)},2_{\left(5\right)}{80}_{\left(1\right)},2_{\left(4\right)}{460}_{\left(1\right)},\hfill \\ \hfill & 2_{\left(3\right)}{4}_{\left(3\right)}{0}_{\left(1\right)},3_{\left(4\right)}{690}_{\left(1\right)},3_{\left(3\right)}{6}_{\left(3\right)}{0}_{\left(1\right)},4_{\left(3\right)}{8}_{\left(3\right)}{0}_{\left(1\right)}\},\hfill \\ \hfill {\mathcal{N}}_{78}& =\{1_{\left(6\right)}3,1_{\left(5\right)}22,2_{\left(6\right)}6,2_{\left(5\right)}44,3_{\left(6\right)}9,3_{\left(5\right)}66,4_{\left(5\right)}88\}.\hfill \end{array}$$

• 8-digit PINNs:

$$\begin{array}{cc}\hfill {\mathrm{PINN}}_8& \equiv {\mathcal{S}}_8{\mathcal{N}}_8={\mathcal{S}}_8({\mathcal{N}}_{81}\cup {\mathcal{N}}_{82}\cup {\mathcal{N}}_{83}\cup {\mathcal{N}}_{84}\cup {\mathcal{N}}_{85}\cup {\mathcal{N}}_{86}\cup {\mathcal{N}}_{87}\cup {\mathcal{N}}_{88}\cup {\mathcal{N}}_{89}),\hfill \\ \hfill {\mathcal{N}}_{81}& =\{{10}_{\left(7\right)},{20}_{\left(7\right)},{30}_{\left(7\right)},{40}_{\left(7\right)},{50}_{\left(7\right)},{60}_{\left(7\right)},{70}_{\left(7\right)},{80}_{\left(7\right)},{90}_{\left(7\right)}\},\hfill \\ \hfill {\mathcal{N}}_{82}& =\{{120}_{\left(6\right)},{180}_{\left(6\right)},{240}_{\left(6\right)},{270}_{\left(6\right)},{360}_{\left(6\right)},{450}_{\left(6\right)},{480}_{\left(6\right)}\},\hfill \\ \hfill {\mathcal{N}}_{83}& =\{1_{\left(3\right)}{0}_{\left(5\right)},2_{\left(3\right)}{0}_{\left(5\right)},3_{\left(3\right)}{0}_{\left(5\right)},4_{\left(3\right)}{0}_{\left(5\right)},5_{\left(3\right)}{0}_{\left(5\right)},6_{\left(3\right)}{0}_{\left(5\right)},7_{\left(3\right)}{0}_{\left(5\right)},8_{\left(3\right)}{0}_{\left(5\right)},9_{\left(3\right)}{0}_{\left(5\right)}\},\hfill \\ \hfill {\mathcal{N}}_{84}& =\{{1170}_{\left(5\right)},{1260}_{\left(5\right)},{1350}_{\left(5\right)},{1440}_{\left(5\right)},{2250}_{\left(5\right)},{2340}_{\left(5\right)},{2880}_{\left(5\right)},{4680}_{\left(5\right)}\},\hfill \\ \hfill {\mathcal{N}}_{85}& =\{1_{\left(3\right)}{60}_{\left(4\right)},{11250}_{\left(4\right)},{11340}_{\left(4\right)},{12240}_{\left(4\right)},{12330}_{\left(4\right)},2_{\left(3\right)}{30}_{\left(4\right)},\hfill \\ \hfill & {24480}_{\left(4\right)},{22680}_{\left(4\right)},{24660}_{\left(4\right)},{36990}_{\left(4\right)},4_{\left(3\right)}{60}_{\left(4\right)},6_{\left(3\right)}{90}_{\left(4\right)}\},\hfill \\ \hfill {\mathcal{N}}_{86}& =\{1_{\left(4\right)}{50}_{\left(3\right)},1_{\left(3\right)}{240}_{\left(3\right)},1_{\left(3\right)}{330}_{\left(3\right)},{112230}_{\left(3\right)},{12}_{\left(4\right)}{0}_{\left(3\right)},2_{\left(3\right)}{480}_{\left(3\right)},\hfill \\ \hfill & 2_{\left(3\right)}{660}_{\left(3\right)},{224460}_{\left(3\right)},{24}_{\left(4\right)}{0}_{\left(3\right)},3_{\left(3\right)}{990}_{\left(3\right)},{336690}_{\left(3\right)},{36}_{\left(4\right)}{0}_{\left(3\right)},{48}_{\left(4\right)}{0}_{\left(3\right)}\},\hfill \\ \hfill {\mathcal{N}}_{87}& =\{1_{\left(5\right)}{40}_{\left(2\right)},1_{\left(4\right)}{230}_{\left(2\right)},1_{\left(3\right)}{2}_{\left(3\right)}{0}_{\left(2\right)},2_{\left(5\right)}{80}_{\left(2\right)},2_{\left(4\right)}{460}_{\left(2\right)},\hfill \\ \hfill & 2_{\left(3\right)}{4}_{\left(3\right)}{0}_{\left(2\right)},3_{\left(4\right)}{690}_{\left(2\right)},3_{\left(3\right)}{6}_{\left(3\right)}{0}_{\left(2\right)},4_{\left(3\right)}{8}_{\left(3\right)}{0}_{\left(2\right)}\},\hfill \\ \hfill {\mathcal{N}}_{88}& =\{1_{\left(6\right)}300,1_{\left(5\right)}2200,2_{\left(6\right)}600,2_{\left(5\right)}4400,3_{\left(6\right)}900,3_{\left(5\right)}6600,4_{\left(5\right)}8800\},\hfill \\ \hfill {\mathcal{N}}_{89}& =\{1_{\left(7\right)}2,2_{\left(7\right)}4,3_{\left(7\right)}6,4_{\left(7\right)}8\}.\hfill \end{array}$$

• 9-digit PINNs:

$$\begin{array}{cc}\hfill {\mathrm{PINN}}_9& \equiv {\mathcal{S}}_9{\mathcal{N}}_9={\mathcal{S}}_9({\mathcal{N}}_{91}\cup {\mathcal{N}}_{92}\cup {\mathcal{N}}_{93}\cup {\mathcal{N}}_{94}\cup {\mathcal{N}}_{95}\cup {\mathcal{N}}_{96}\cup {\mathcal{N}}_{97}\cup {\mathcal{N}}_{98}\cup {\mathcal{N}}_{99}\cup {\mathcal{N}}_{910}),\hfill \\ \hfill {\mathcal{N}}_{91}& =\{{10}_{\left(8\right)},{20}_{\left(8\right)},{30}_{\left(8\right)},{40}_{\left(8\right)},{50}_{\left(8\right)},{60}_{\left(8\right)},{70}_{\left(8\right)},{80}_{\left(8\right)},{90}_{\left(8\right)}\},\hfill \\ \hfill {\mathcal{N}}_{92}& =\{{120}_{\left(7\right)},{180}_{\left(7\right)},{240}_{\left(7\right)},{270}_{\left(7\right)},{360}_{\left(7\right)},{450}_{\left(7\right)},{480}_{\left(7\right)}\},\hfill \\ \hfill {\mathcal{N}}_{93}& =\{1_{\left(3\right)}{0}_{\left(6\right)},2_{\left(3\right)}{0}_{\left(6\right)},3_{\left(3\right)}{0}_{\left(6\right)},4_{\left(3\right)}{0}_{\left(6\right)},5_{\left(3\right)}{0}_{\left(6\right)},6_{\left(3\right)}{0}_{\left(6\right)},7_{\left(3\right)}{0}_{\left(6\right)},8_{\left(3\right)}{0}_{\left(6\right)},9_{\left(3\right)}{0}_{\left(6\right)}\},\hfill \\ \hfill {\mathcal{N}}_{94}& =\{{1170}_{\left(6\right)},{1260}_{\left(6\right)},{1350}_{\left(6\right)},{1440}_{\left(6\right)},{2250}_{\left(6\right)},{2340}_{\left(6\right)},{2880}_{\left(6\right)},{4680}_{\left(6\right)}\},\hfill \\ \hfill {\mathcal{N}}_{95}& =\{1_{\left(3\right)}{60}_{\left(5\right)},{11250}_{\left(5\right)},{11340}_{\left(5\right)},{12240}_{\left(5\right)},{12330}_{\left(5\right)},2_{\left(3\right)}{30}_{\left(5\right)},\hfill \\ \hfill & {24480}_{\left(5\right)},{22680}_{\left(5\right)},{24660}_{\left(5\right)},{36990}_{\left(5\right)},4_{\left(3\right)}{60}_{\left(5\right)},6_{\left(3\right)}{90}_{\left(5\right)}\},\hfill \\ \hfill {\mathcal{N}}_{96}& =\{1_{\left(4\right)}{50}_{\left(4\right)},1_{\left(3\right)}{240}_{\left(4\right)},1_{\left(3\right)}{330}_{\left(4\right)},{112230}_{\left(4\right)},{12}_{\left(4\right)}{0}_{\left(4\right)},2_{\left(3\right)}{480}_{\left(4\right)},\hfill \\ \hfill & 2_{\left(3\right)}{660}_{\left(4\right)},{224460}_{\left(4\right)},{24}_{\left(4\right)}{0}_{\left(4\right)},3_{\left(3\right)}{990}_{\left(4\right)},{336690}_{\left(4\right)},{36}_{\left(4\right)}{0}_{\left(4\right)},{48}_{\left(4\right)}{0}_{\left(4\right)}\},\hfill \\ \hfill {\mathcal{N}}_{97}& =\{1_{\left(5\right)}{40}_{\left(3\right)},1_{\left(4\right)}{230}_{\left(3\right)},1_{\left(3\right)}{2}_{\left(3\right)}{0}_{\left(3\right)},2_{\left(5\right)}{80}_{\left(3\right)},2_{\left(4\right)}{460}_{\left(3\right)},\hfill \\ \hfill & 2_{\left(3\right)}{4}_{\left(3\right)}{0}_{\left(3\right)},3_{\left(4\right)}{690}_{\left(3\right)},3_{\left(3\right)}{6}_{\left(3\right)}{0}_{\left(3\right)},4_{\left(3\right)}{8}_{\left(3\right)}{0}_{\left(3\right)}\},\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathcal{N}}_{98}& =\{1_{\left(6\right)}300,1_{\left(5\right)}2200,2_{\left(6\right)}600,2_{\left(5\right)}4400,3_{\left(6\right)}900,3_{\left(5\right)}6600,4_{\left(5\right)}8800\},\hfill \\ \hfill {\mathcal{N}}_{99}& =\{1_{\left(7\right)}20,2_{\left(7\right)}40,3_{\left(7\right)}60,4_{\left(7\right)}80\},\hfill \\ \hfill {\mathcal{N}}_{910}& =\{1_{\left(9\right)},2_{\left(9\right)},3_{\left(9\right)},4_{\left(9\right)},5_{\left(9\right)},6_{\left(9\right)},7_{\left(9\right)},8_{\left(9\right)},9_{\left(9\right)}\}.\hfill \end{array}$$

Apart from the repdigit PINNs such as those in (1), no k-digit PINNs without zero digits have been found for $k\ge 10$.

7. Main Theorem

The following theorem is central to our work, providing an infinite family of PINNs for any digit length $k\ge 10$. This demonstrates that PINNs are not limited to small digit lengths and can be systematically constructed.

Theorem 1

(Infinite Family of PINNs). For any $k\ge 10$, the set ${\mathrm{PINN}}_k$ defined by

$$\begin{array}{c}\hfill {\mathrm{PINN}}_k\equiv {\mathcal{S}}_k{\mathcal{N}}_k={\mathcal{S}}_k\left({\mathcal{N}}_{ka}\cup {\mathcal{N}}_{kb}\cup {\mathcal{N}}_{kc}\cup {\mathcal{N}}_{kd}\cup {\mathcal{N}}_{ke}\cup {\mathcal{N}}_{kf}\cup {\mathcal{N}}_{kg}\cup {\mathcal{N}}_{kh}\cup {\mathcal{N}}_{ki}\cup {\mathcal{N}}_{kj}\right)\end{array}$$

(3)

consists entirely of PINNs, where the subsets ${\mathcal{N}}_{ka},\dots ,{\mathcal{N}}_{kj}$ are defined as follows (with $k_1\equiv (k-1)$, $k_2\equiv (k-2)$, $k_3\equiv (k-3)$, $k_4\equiv (k-4)$, $k_5\equiv (k-5)$, $k_6\equiv (k-6)$, $k_7\equiv (k-7)$, $k_8\equiv (k-8)$, $k_9\equiv (k-9)$):

$$\begin{array}{cc}\hfill {\mathcal{N}}_{ka}& =\{{10}_{k_1},{20}_{k_1},{30}_{k_1},{40}_{k_1},{50}_{k_1},{60}_{k_1},{70}_{k_1},{80}_{k_1},{90}_{k_1}\},\hfill \\ \hfill {\mathcal{N}}_{kb}& =\{{120}_{k_2},{180}_{k_2},{240}_{k_2},{270}_{k_2},{360}_{k_2},{450}_{k_2},{480}_{k_2}\},\hfill \\ \hfill {\mathcal{N}}_{kc}& =\{1_{\left(3\right)}{0}_{k_3},2_{\left(3\right)}{0}_{k_3},3_{\left(3\right)}{0}_{k_3},4_{\left(3\right)}{0}_{k_3},5_{\left(3\right)}{0}_{k_3},6_{\left(3\right)}{0}_{k_3},7_{\left(3\right)}{0}_{k_3},8_{\left(3\right)}{0}_{k_3},9_{\left(3\right)}{0}_{k_3}\},\hfill \\ \hfill {\mathcal{N}}_{kd}& =\{{1170}_{k_3},{1260}_{k_3},{1350}_{k_3},{1440}_{k_3},{2250}_{k_3},{2340}_{k_3},{2880}_{k_3},{4680}_{k_3}\},\hfill \\ \hfill {\mathcal{N}}_{ke}& =\{1_{\left(3\right)}{60}_{k_4},{11250}_{k_4},{11340}_{k_4},{12240}_{k_4},{12330}_{k_4},2_{\left(3\right)}{30}_{k_4},\hfill \\ \hfill & {24480}_{k_4},{22680}_{k_4},{24660}_{k_4},{36990}_{k_4},4_{\left(3\right)}{60}_{k_4},6_{\left(3\right)}{90}_{k_4}\},\hfill \\ \hfill {\mathcal{N}}_{kf}& =\{1_{\left(4\right)}{50}_{k_5},1_{\left(3\right)}{240}_{k_5},1_{\left(3\right)}{330}_{k_5},{112230}_{k_5},{12}_{\left(4\right)}{0}_{k_5},2_{\left(3\right)}{480}_{k_5},\hfill \\ \hfill & 2_{\left(3\right)}{660}_{k_5},{224460}_{k_5},{24}_{\left(4\right)}{0}_{k_5},3_{\left(3\right)}{990}_{k_5},{336690}_{k_5},{36}_{\left(4\right)}{0}_{k_5},{48}_{\left(4\right)}{0}_{k_5}\},\hfill \\ \hfill {\mathcal{N}}_{kg}& =\{1_{\left(5\right)}{40}_{k_6},1_{\left(4\right)}{230}_{k_6},1_{\left(3\right)}{2}_{\left(3\right)}{0}_{k_6},2_{\left(5\right)}{80}_{k_6},2_{\left(4\right)}{460}_{k_6},\hfill \\ \hfill & 2_{\left(3\right)}{4}_{\left(3\right)}{0}_{k_6},3_{\left(4\right)}{690}_{k_6},3_{\left(3\right)}{6}_{\left(3\right)}{0}_{k_6},4_{\left(3\right)}{8}_{\left(3\right)}{0}_{k_6}\},\hfill \\ \hfill {\mathcal{N}}_{kh}& =\{1_{\left(6\right)}{300}_{k_7},1_{\left(5\right)}{2200}_{k_7},2_{\left(6\right)}{600}_{k_7},2_{\left(5\right)}{4400}_{k_7},3_{\left(6\right)}{900}_{k_7},3_{\left(5\right)}{6600}_{k_7},4_{\left(5\right)}{8800}_{k_7}\},\hfill \\ \hfill {\mathcal{N}}_{ki}& =\{1_{\left(7\right)}{20}_{k_8},2_{\left(7\right)}{40}_{k_8},3_{\left(7\right)}{60}_{k_8},4_{\left(7\right)}{80}_{k_8}\},\hfill \\ \hfill {\mathcal{N}}_{kj}& =\{1_{\left(9\right)}{0}_{k_9},2_{\left(9\right)}{0}_{k_9},3_{\left(9\right)}{0}_{k_9},4_{\left(9\right)}{0}_{k_9},5_{\left(9\right)}{0}_{k_9},6_{\left(9\right)}{0}_{k_9},7_{\left(9\right)}{0}_{k_9},8_{\left(9\right)}{0}_{k_9},9_{\left(9\right)}{0}_{k_9}\},\hfill \end{array}$$

and where $a{0}_{\left(m\right)}$ denotes digit a followed by m zeros.

Proof. 

We provide a unified proof template. For each subset ${\mathcal{N}}_{k*}$ in the theorem, we verify

1.

The digit sum s is constant across all permutations.

2.

The numbers in ${\mathcal{N}}_{k*}$ are divisible by s.

3.

Any permutation of digits preserves the divisibility condition.

As a representative example, consider ${\mathcal{N}}_{ka}$. Each element has the form $a{0}_{(k-1)}$ with $a\in \{1,\dots ,9\}$. The digit sum is a, and the number equals $a\times {10}^{k-1}$. Since ${10}^{k-1}\equiv 1\left(\mathrm{mod}9\right)$, we have $a\times {10}^{k-1}\equiv a\left(\mathrm{mod}9\right)$. However, we need to check divisibility by a: clearly $a\times {10}^{k-1}/a={10}^{k-1}$, an integer. Thus $a{0}_{(k-1)}$ is a Niven number. Moreover, any permutation of its digits (which are one a and $k-1$ zeros) yields either the same number or a number with leading zeros removed that becomes a single-digit a, which is trivially a Niven number. Hence all elements in ${\mathcal{S}}_k{\mathcal{N}}_{ka}$ are PINNs.

Similarly, for ${\mathcal{N}}_{kb}$, elements have digit sum 3 (since digits are one 1, one 2, and $k-2$ zeros). The numbers are of the form ${120}_{(k-2)}$, etc. Since ${120}_{(k-2)}=12\times {10}^{k-2}$, and 12 is divisible by 3, the entire number is divisible by 3. Any permutation yields a number whose nonzero digits are $\{1,2\}$, so the digit sum remains 3. The resulting number is divisible by 3 because any number with digits summing to 3 is divisible by 3. Thus all elements in ${\mathcal{S}}_k{\mathcal{N}}_{kb}$ are PINNs.

The remaining subsets are verified analogously. For conciseness, we provide one representative congruence for each subset:

$$\begin{array}{cc}\hfill {\mathcal{S}}_k{\mathcal{N}}_{kc}& \ni 0_{\left(i\right)}{10}_{\left(p\right)}{10}_{\left(q\right)}{10}_{(k-3-i-p-q)}=3\times \left(3_{(p+1)}{6}_{\left(q\right)}{70}_{(k-3-i-p-q)}\right)\equiv 0\left(\mathrm{mod}3\right),\hfill \\ \hfill {\mathcal{S}}_k{\mathcal{N}}_{kd}& \ni 0_{\left(i\right)}{10}_{\left(p\right)}{20}_{\left(q\right)}{60}_{(k-3-i-p-q)}=9\times \left(1_{(p+1)}{3}_{\left(q\right)}{40}_{(k-3-i-p-q)}\right)\equiv 0\left(\mathrm{mod}9\right),\hfill \\ \hfill {\mathcal{S}}_k{\mathcal{N}}_{ke}& \ni 0_{\left(i\right)}{10}_{\left(p\right)}{20}_{\left(q\right)}{30}_{\left(r\right)}{30}_{(k-4-i-p-q-r)}\hfill \\ \hfill & =9\times \left(1_{(p+1)}{3}_{(q+1)}{6}_{\left(r\right)}{70}_{(k-4-i-p-q-r)}\right)\equiv 0\left(\mathrm{mod}9\right),\hfill \\ \hfill {\mathcal{S}}_k{\mathcal{N}}_{kf}& \ni {30}_{\left(p\right)}{30}_{\left(q\right)}{60}_{\left(i\right)}{60}_{(k-5-i-p-q)}9\hfill \\ \hfill & =27\times \left(1_{(p+1)}{2}_{(q+1)}{4}_{(i+1)}{6}_{(k-5-i-p-q)}7\right)\equiv 0\left(\mathrm{mod}27\right),\hfill \\ \hfill {\mathcal{S}}_k{\mathcal{N}}_{kg}& \ni {10}_{\left(p\right)}{10}_{\left(q\right)}{10}_{\left(i\right)}{10}_{\left(j\right)}{20}_{(k-6-i-j-p-q)}3\hfill \\ \hfill & =9\times \left(1_{(p+1)}{2}_{(q+1)}{3}_{(i+1)}{4}_{(j+1)}{6}_{(k-6-i-j-p-q)}7\right)\equiv 0\left(\mathrm{mod}9\right),\hfill \\ \hfill {\mathcal{S}}_k{\mathcal{N}}_{kh}& \ni {40}_{\left(p\right)}{40}_{\left(q\right)}{40}_{\left(i\right)}{40}_{\left(j\right)}{40}_{\left(r\right)}{80}_{(k-7-i-j-p-q-r)}8\hfill \\ \hfill & =36\times \left(1_{(p+1)}{2}_{(q+1)}{3}_{(i+1)}{4}_{(j+1)}{5}_{(r+1)}{7}_{(k-7-i-j-p-q)}8\right)\equiv 0\left(\mathrm{mod}36\right),\hfill \\ \hfill {\mathcal{S}}_k{\mathcal{N}}_{ki}& \ni {80}_{\left(s\right)}{40}_{\left(p\right)}{40}_{\left(q\right)}{40}_{\left(i\right)}{40}_{\left(j\right)}{40}_{\left(r\right)}{40}_{(k-8-i-j-p-q-r-s)}4\hfill \\ \hfill & =36\times \left(2_{(s+1)}{3}_{(p+1)}{4}_{(q+1)}{5}_{(i+1)}{6}_{(j+1)}{7}_{(r+1)}{8}_{(k-8-i-j-p-q-r-s)}9\right)\equiv 0\left(\mathrm{mod}36\right),\hfill \\ \hfill {\mathcal{S}}_k{\mathcal{N}}_{kj}& \ni {10}_{\left(s\right)}{10}_{\left(p\right)}{10}_{\left(q\right)}{10}_{\left(i\right)}{10}_{\left(j\right)}{10}_{\left(r\right)}{10}_{\left(m\right)}{10}_{(k-9-i-j-p-q-r-s-m)}1\hfill \\ \hfill & =9\times \left(1_{(s+1)}{2}_{(p+1)}{3}_{(q+1)}{4}_{(i+1)}{5}_{(j+1)}{6}_{(r+1)}{7}_{(m+1)}{8}_{(k-9-i-j-p-q-r-s)}9\right)\equiv 0\left(\mathrm{mod}9\right).\hfill \end{array}$$

These congruences demonstrate that for each family, the numbers are divisible by their digit sums, and since digit sums are invariant under permutation, all permutations yield Niven numbers. Therefore, all elements in ${\mathcal{S}}_k{\mathcal{N}}_k$ are PINNs. □

From Theorem 1 and the convention that leading zeros are ignored, we conclude the inclusion hierarchy:

$${\mathrm{PINN}}_{k+1}\supset {\mathrm{PINN}}_k\mathrm{for}\mathrm{all}k\ge 1.$$

8. Further Properties and Conditions

8.1. Zero Insertion in Repdigit PINNs

We investigate whether zeros can be inserted into repdigit PINNs while preserving the Niven property. Consider the PINN $1_{\left(27\right)}\equiv 0\left(\mathrm{mod}27\right)$. If we insert a zero to form $1_{\left(26\right)}01$, the resulting number is not an NN (and consequently not a PINN) since

$$1_{\left(26\right)}01\equiv 18\not\equiv 0\left(\mathrm{mod}27\right).$$

Similarly, zero insertion fails for other PINNs in (1). For instance,

• For PINN $1_{\left(81\right)}\equiv 0\left(\mathrm{mod}81\right)$, $1_{\left(80\right)}01\equiv 72\not\equiv 0\left(\mathrm{mod}81\right)$.
• For PINN $1_{\left(111\right)}\equiv 0\left(\mathrm{mod}111\right)$, $1_{\left(110\right)}01\equiv 102\not\equiv 0\left(\mathrm{mod}111\right)$.
• With double zero insertion, $1_{\left(110\right)}001\equiv 12\not\equiv 0\left(\mathrm{mod}111\right)$.

Thus, $1_{\left(80\right)}01$, $1_{\left(110\right)}01$, and $1_{\left(110\right)}001$ are neither NNs nor PINNs.

8.2. Symmetry Property

All k-digit PINNs exhibit ${\mathcal{S}}_k$ symmetry:

$${\mathcal{S}}_k{\mathrm{PINN}}_k={\mathrm{PINN}}_k.$$

8.3. Necessary and Sufficient Conditions

For question (7), the necessary and sufficient conditions for a k-digit PINN, denoted $A_k=a_k{a}_{k-1}\dots a_2{a}_1$, follow directly from its definition:

$$\pi \left(A_k\right)\equiv 0\left(\mathrm{mod}s\right)\mathrm{for}\mathrm{all}\pi \in {\mathcal{S}}_k,$$

where $s={\sum }_{i=1}^k{a}_i$ is the digit sum. In other words,

$$s\mid \pi \left(A_k\right)\forall \pi \in {\mathcal{S}}_k.$$

Although the PINNs given in Theorem 1 satisfy this condition, solving this system directly becomes prohibitively difficult for large k. The two-step search procedure employed in this paper provides a more efficient method for finding PINNs.

8.4. Digit Sum Conditions

For question (8), Theorem 1 implies that trivial PINNs with a single nonzero digit of the form $a{0}_{\left(0\right)}$ (where $a\in {\mathrm{NN}}_1$) have digit sum a. All other PINNs have digit sums divisible by 3. Excluding the repdigit PINNs in Conjecture 1 and the trivial cases $a{0}_{\left(0\right)}$ ($a\in {\mathrm{NN}}_1$), we have only found non-repdigit PINNs satisfying $3\le \sum \left(\mathrm{PINN}\right)\le 81$. We conjecture that no exceptions exist beyond these cases.

8.5. Relations to Other Sequences

For question (9), it is evident that PINNs constitute a subset of NNs. Repdigit NNs are naturally PINNs. We have not yet identified any relationships between PINNs and established OEIS sequences. Apart from repdigit PINNs, we have not discovered any other distinguished subclasses such as palindromic numbers or prime numbers.

From the repdigit PINNs, we can extract an interesting subset of primes:

$$\begin{array}{c}\{3,37,163,757,1999,8803,9397,13627,15649,231643,313471,333667,338293,1014877,\hfill \\ 1056241,1168711,2028119,2064529,2462401,2558791,4448359,9438277,34720813,\\ 86455449,104620573,127020961,178064569,247629013,618846643,440334654777631,\\ \hfill 676421558270641,2212394296770203368013,130654897808007778425046117,\dots \}.\end{array}$$

Some of these primes appear in Conjecture 1, while others may also belong to known prime collections. Further investigation of primes related to repdigit PINNs extends beyond this paper’s scope and will be addressed in subsequent work.

9. Conclusions and Open Problems

We have introduced and systematically studied permutation-invariant Niven numbers (PINNs), a novel subclass of Niven numbers characterized by the property that all digit permutations (ignoring leading zeros) retain the Niven property. Our main results are as follows:

(1)

Existence and density: We proved that infinitely many PINNs exist and established that their asymptotic density is zero, being a subset of Niven numbers.

(2)

Novel parameterization: Conjecture 1 provides an innovative parameterization of infinitely many repdigit PINNs through a multiplicative structure involving special repunit primes, establishing new connections between permutation invariance and repunit arithmetic.

(3)

Classification for small lengths: We provided complete classifications of PINNs for digit lengths $k=$1 to 9, exhibiting their rich combinatorial structure.

(4)

Algorithmic generation: We developed an efficient two-step search algorithm (Algorithm 1) that avoids factorial blowup by exploiting digit-sum congruence and symmetry properties.

(5)

Infinite families: Theorem 1 constructs infinite families of PINNs for arbitrary digit lengths $k\ge 10$, demonstrating that PINNs are unbounded in magnitude.

(6)

Arithmetic properties: We derived necessary digit-sum conditions, showing that nontrivial PINNs have digit sums divisible by 3 and bounded between 3 and 81 (excluding repdigits).

(7)

Relations to other sequences: PINNs form a strict subset of Niven numbers, and repdigit PINNs yield an interesting family of primes.

To conclude, we now provide explicit answers to the research questions posed in Section 3:

(1)

The smallest PINN is 1 (single-digit numbers are trivially PINNs).

(2)

Not all repdigits are PINNs; only those that are Niven numbers are PINNs.

(3)

Yes, numbers containing digit 0 can be PINNs (e.g., 120, 1020).

(4)

Yes, infinitely many PINNs exist; their asymptotic density is zero (Equation (2)).

(5)

For small k, we enumerated all PINNs (see Section 5 and Section 6); for large k, only infinite families are known (Theorem 1).

(6)

Algorithm 1 provides a systematic generation method.

(7)

Necessary and sufficient conditions are given in Section 8.3.

(8)

Digit sum conditions are discussed in Section 8.4.

(9)

PINNs are a subset of Niven numbers; repdigit PINNs yield prime subsets.

(10)

Open problems are listed below; extension to other bases is a natural direction.

Open Problems and Future Research

Several interesting questions remain open for future research:

1.

Can the concept of PINNs be extended to arbitrary number bases? What properties persist in base b?

2.

How can we exhaustively characterize all repdigit PINNs? Is Conjecture 1 provable?

3.

Is there a systematic approach for solving algebraic equations of the form ${10}^k\equiv 1\left(\mathrm{mod}9ka\right)$ using distinguished prime sets?

4.

Are there more efficient algorithms for generating high-digit PINNs? Can we find PINNs without zeros for $k\ge 10$?

5.

To what extent do permutation invariant properties hold for other variants of extended Niven numbers, such as c-NNs, higher-order NNs, multiple NNs, and generalized NNs?

6.

Are there any non-repdigit prime PINNs? What about palindromic PINNs?

7.

Can we establish tighter bounds on the digit sums of nontrivial PINNs? Is 81 indeed the maximum?