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Which number theory topics appear in Books VII through IX of Euclid’s Elements?

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Three theorems of congruent triangles — Euclidean geometry This diagram depicts three basic theorems that triangles are congruent—that is, they are of equal shape and size—if: (1) two sides and the included angle are equal, designated SAS; (2) two angles and the included side are equal, designated ASA; or (3) all three sides are equal, designated SSS.

Books VII through IX of Euclid’s Elements shift focus from geometry to number theory, specifically concerning positive integers greater than 1. These books establish the foundation for number theory as a rigorous mathematical enterprise through several key topics:1

Definitions: The books begin with 22 definitions, including concepts such as unity, even and odd numbers, prime numbers, composite numbers, and perfect numbers.12
Euclidean Algorithm: Book VII introduces antanaresis, now known as the Euclidean algorithm, a method for determining the greatest common divisor of two or more numbers.12
Unique Factorization: Euclid explores the concept of fundamental theorem of arithmetic, which states that every whole number can be factored into a unique product of primes.2
Geometric Sequences: Book VIII examines numbers in continued proportions, which are now referred to as geometric sequences.1
Infinite Primes: Book IX contains a famous proof demonstrating that there is an infinite number of primes.12
Perfect Numbers: Book IX concludes with a formula for generating perfect numbers based on the sum of powers of 2.2