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Mind the (multiplicative) gaps

Emmanuel Kowalski, Vivian Kuperberg

TL;DR

The paper determines an exact, finite classification of primitive gaps in the NΓ—N multiplication table by partitioning them into three disjoint sets 𝔖_N, 𝔅_N, and π”ˆ_N, with R marking a key threshold derived from ⌊j^2/4βŒ‹<N. The authors develop both a constructive, interval-based method and an asymptotic-polynomial framework to locate gaps, and they prove an exhaustion via induction on N, including several case analyses tied to when π”ˆ_N is empty or nonempty and how R changes to R'. The result reveals a rigid, algebraic layer governing the large gaps and isolates them from the rest, while also detailing multiplicities and providing explicit examples such as N=33 and N=100. This advances understanding of the image size and gap distribution in the multiplication table beyond probabilistic heuristics by illuminating precise structural constraints.

Abstract

We determine the complete list of the gaps between successive elements of the multiplication table of the first N integers.

Mind the (multiplicative) gaps

TL;DR

The paper determines an exact, finite classification of primitive gaps in the NΓ—N multiplication table by partitioning them into three disjoint sets 𝔖_N, 𝔅_N, and π”ˆ_N, with R marking a key threshold derived from ⌊j^2/4βŒ‹<N. The authors develop both a constructive, interval-based method and an asymptotic-polynomial framework to locate gaps, and they prove an exhaustion via induction on N, including several case analyses tied to when π”ˆ_N is empty or nonempty and how R changes to R'. The result reveals a rigid, algebraic layer governing the large gaps and isolates them from the rest, while also detailing multiplicities and providing explicit examples such as N=33 and N=100. This advances understanding of the image size and gap distribution in the multiplication table beyond probabilistic heuristics by illuminating precise structural constraints.

Abstract

We determine the complete list of the gaps between successive elements of the multiplication table of the first N integers.
Paper Structure (6 sections, 12 theorems, 112 equations)

This paper contains 6 sections, 12 theorems, 112 equations.

Table of Contents

  1. The gaps in the multiplication table
  2. Examples
  3. The "asymptotic" multiplication table
  4. Disjointness
  5. Finding the gaps
  6. Exhaustion

Key Result

Theorem 1.1

Let $N\geqslant 2$ be an integer. Let $R$ be the largest integer $j$ such that $\lfloor j^2/4\rfloor<N$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Example 3.4
  • ...and 16 more