On the last digits of consecutive primes

TL;DR

This paper models the distribution of last digits of consecutive primes using dynamic systems based on Eratosthenes sieve, revealing biases and their eventual reversal at large primes, complementing probabilistic theories.

Contribution

It introduces exact dynamic systems modeling prime gap populations during Eratosthenes sieve, offering new insights into last digit biases and their evolution.

Findings

Biases in last digits are predicted to reverse for large primes

Exact models describe the evolution of prime gap populations

Provides time constants for gap population changes

Abstract

Recently Oliver and Soundararajan made conjectures based on computational enumerations about the frequency of occurrence of pairs of last digits for consecutive primes. By studying Eratosthenes sieve, we have identified discrete dynamic systems that exactly model the populations of gaps across stages of Eratosthenes sieve. Our models provide some insight into the observed biases in the occurrences of last digits in consecutive primes, and the models suggest that the biases will ultimately be reversed for large enough primes. The exact model for populations of gaps across stages of Eratosthenes sieve provides a constructive complement to the probabilistic models rooted in the work of Hardy and Littlewood, and it provides time constants that describe the evolution of the populations of larger gaps.