On the probability of co-primality of two natural numbers chosen at random (Who was the first to pose and solve this problem?)

TL;DR

This paper explores the historical development and various mathematical approaches to determining the probability that two randomly chosen natural numbers are coprime, highlighting its significance in probability and number theory.

Contribution

It provides a comprehensive historical overview and analysis of different methods and generalizations related to the coprimality probability problem.

Findings

Historical origins of the problem identified

Multiple approaches to calculating coprimality probability reviewed

Various generalizations and related problems discussed

Abstract

The article attempts to demonstrate the rich history of one truly remarkable problem situated at the confluence of probability theory and theory of numbers - finding the probability of co-primality of two randomly selected natural numbers. The goal of the article is to reveal the genesis of the problem and understand the role of mathematicians of the past in solving this problem and using it as a background for advancing mathematical ideas. The article describes different approaches to the solution of the problem, reviews its various generalizations, examines similar problems, and offers diverse historical perspectives.