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Pandigital and penholodigital numbers

Chai Wah Wu

TL;DR

The paper investigates base-$b$ pandigital and penholodigital numbers, including strict and subvariants, and their intersections with squares, oblongs, and primes. It develops a modular framework built on the digit-sum identity $s_b(n)$ and residue sets such as $A_b$ and $B_b$, yielding precise nonexistence results in many bases and explicit lower bounds for prime and square-type numbers. Notably, it shows that strict pandigital/penholodigital primes are absent for bases $b>3$, provides lower bounds and conjectures for smallest primes, and extends the analysis to subpandigital and subpenholodigital families with analogous bounds and OEIS references. These results illuminate how digit-structure constraints interact with classical number-theoretic objects, guiding computation and further theoretical exploration across bases.

Abstract

Pandigital and penholodigital numbers are numbers that contain every digit or nonzero digit respectively. We study properties of pandigital or penholodigital numbers that are also square, oblong or prime.

Pandigital and penholodigital numbers

TL;DR

The paper investigates base- pandigital and penholodigital numbers, including strict and subvariants, and their intersections with squares, oblongs, and primes. It develops a modular framework built on the digit-sum identity and residue sets such as and , yielding precise nonexistence results in many bases and explicit lower bounds for prime and square-type numbers. Notably, it shows that strict pandigital/penholodigital primes are absent for bases , provides lower bounds and conjectures for smallest primes, and extends the analysis to subpandigital and subpenholodigital families with analogous bounds and OEIS references. These results illuminate how digit-structure constraints interact with classical number-theoretic objects, guiding computation and further theoretical exploration across bases.

Abstract

Pandigital and penholodigital numbers are numbers that contain every digit or nonzero digit respectively. We study properties of pandigital or penholodigital numbers that are also square, oblong or prime.
Paper Structure (6 sections, 16 theorems, 1 table)

This paper contains 6 sections, 16 theorems, 1 table.

Table of Contents

  1. Introduction
  2. Pandigital and penholodigital squares
  3. Pandigital and penholodigital oblong numbers
  4. Pandigital and penholodigital primes
  5. Subpandigital and subpenholodigital numbers
  6. Conclusions

Key Result

Theorem 1

Let $A_b$ be the set of modular square roots of $b(b-1)/2$ modulo $b-1$, i.e. it is the set of integers $0\leq m < b-1$ such that $m^2 \equiv b(b-1)/2 \pmod{b-1}$. If $n^2$ is a strict pandigital or a strict penholodigital square, then $n\equiv m \pmod{b-1}$ for some $m\in A_b$.

Theorems & Definitions (31)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • proof
  • Corollary 1
  • Conjecture 1
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • ...and 21 more