On the Success of Mishandling Euclid's Lemma

TL;DR

This paper investigates the frequent misapplication of Euclid's lemma by students to non-prime numbers, demonstrating that such implications are almost always false and providing a probabilistic analysis.

Contribution

It provides a probabilistic framework quantifying the likelihood of incorrect applications of Euclid's lemma to composite numbers.

Findings

Misapplication of Euclid's lemma to non-primes is almost surely false.

Provides an asymptotic formula for the probability of such misapplications.

Highlights the importance of correct prime-based reasoning in number theory.

Abstract

We examine Euclid's lemma that if $p$ is a prime number such that $p | ab$, then $p$ divides at least one of $a$ or $b$. Specifically, we consider the common misapplication of this lemma to numbers that are not prime, as is often made by undergraduate students. We show that a randomly chosen implication of the form $r |ab \Rightarrow r|a \text{ or } r|b$ is almost surely false in a probabilistic sense, and we quantify this with a corresponding asymptotic formula.