Distributions of left prime truncations

TL;DR

This paper investigates the distribution of left prime truncations in integers and irreducible truncations in polynomials over finite fields, analyzing their proportions, variance, and maximums to understand their statistical properties.

Contribution

It introduces a comprehensive analysis of the distribution patterns of left prime truncations and irreducible polynomial truncations, highlighting their statistical behavior.

Findings

Distribution of left prime truncations varies with number length.

Proportions of irreducible polynomial truncations are characterized.

Variance and maximum proportions are quantified.

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.