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Distributions of left prime truncations

Vivian Kuperberg, Matilde Lalín

Abstract

The prime number 357686312646216567629137 is notable because of the unusual property that it remains prime successively on removing the left digit until there are no remaining digits. We explore here the distributions of the number of left prime truncations of integers and of the number of irreducible truncations of polynomials with coefficients over a finite field, focusing on the proportion among all $\ell$-digit numbers or polynomials, their variance, and the maximal proportion.

Distributions of left prime truncations

Abstract

The prime number 357686312646216567629137 is notable because of the unusual property that it remains prime successively on removing the left digit until there are no remaining digits. We explore here the distributions of the number of left prime truncations of integers and of the number of irreducible truncations of polynomials with coefficients over a finite field, focusing on the proportion among all -digit numbers or polynomials, their variance, and the maximal proportion.
Paper Structure (7 sections, 12 theorems, 130 equations)

This paper contains 7 sections, 12 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. The integer setting
  3. Proof of Lemma \ref{['lem:11']}
  4. A heuristic for the maximum number of prime truncations in the integer setting
  5. The polynomial setting
  6. Proof of Lemma \ref{['lem:correlations-using-random-matrix-theory-general']}
  7. A heuristic for the maximum number of irreducible truncations of a polynomial

Key Result

Theorem 1.6

As $b\rightarrow \infty$, we have and as $\ell\rightarrow \infty$, we have

Theorems & Definitions (26)

  • Definition 1.1
  • Definition 1.3
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Conjecture 1.9
  • Theorem 1.10
  • Conjecture 1.11
  • Lemma 2.1
  • ...and 16 more