Introduction to number theory lecture 12 Wilson's theorem

Item: Introduction to number theory lecture 12 Wilson's theorem
Rating: 5
Author: PostSingularity Institute

by Richard E Borcherds

Formal & Physical Sciences Mathematics PBMathematics PBHNumber theory

🎙 Richard E Borcherds 👥 82K 📅 February 1, 2022 ⏱ 29 min 👁 9K 📄 lecture 🧭 2026-08-17

Available in: English (current) Français

Keywords

Wilson's theoremfactorialprimemodular arithmeticsquare root

Summary

This lecture from the Berkeley Math 115 course introduces Wilson’s theorem, which states that for a prime p, (p-1)! ≡ -1 mod p. The lecturer begins by computing (m-1)! modulo m for small m, observing the pattern for primes. He then proves the theorem by pairing each number from 2 to p-2 with its multiplicative inverse modulo p, leaving only 1 and -1. He also discusses the converse: if (m-1)! ≡ -1 mod m, then m is prime, but notes this is not a practical primality test. The lecture then explores applications: finding square roots of -1 modulo p, proving Fermat’s little theorem and Euler’s theorem via product arguments, and analyzing the product of all units modulo m, which leads to a generalization of Wilson’s theorem. The lecturer also discusses the case where the product is 1 instead of -1, such as for m=8 and m=15, and explains the role of solutions to x^2 ≡ 1 mod m.

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Critical Evaluation

Value of the Information & Strength of the Argument

The lecture provides a clear and rigorous exposition of Wilson’s theorem and its applications. The argumentation is solid, with step-by-step proofs and illustrative examples. The lecturer also highlights the limitations of the theorem as a primality test and discusses related results, such as the product of units modulo m, which enriches the content. The value lies in the pedagogical approach and the connections made between different concepts in number theory.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is based on the textbook ‘An Introduction to the Theory of Numbers’ by Niven, Zuckerman, and Montgomery, which is a standard reference. The mathematical content is accurate and well-presented. The title accurately reflects the content, as the lecture focuses on Wilson’s theorem and its applications. No external sources are cited beyond the textbook and the course playlist.

144 words

Title / Content Match

The title accurately reflects the content, which is a focused lecture on Wilson's theorem and its applications.

Quality & Reliability

9/10

The lecture is given by a renowned mathematician (Richard Borcherds, Fields Medalist) and presents rigorous proofs and examples. The content is mathematically sound and well-structured.

Key Moments

Markers derived by PSI from the transcript: the creator did not define chapters.

Cited Sources

Course playlist — Other lectures in the course

Concurring Sources

An Introduction to the Theory of Numbers — Textbook used for the course

Contribution & Novelties

The lecture offers a clear and thorough explanation of Wilson’s theorem, including its proof and applications. It also provides insights into related results, such as the product of units modulo m, which is a generalization. The lecturer’s approach of pairing inverses is particularly illuminating.

Pour aller plus loin :

Wilson’s theorem - Wikipedia — Background and history.
Fermat’s little theorem - Wikipedia — Related theorem.
Euler’s theorem - Wikipedia — Generalization of Fermat’s little theorem.

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Radar Profile

The radar profile shows high scores in information quantity, quality, and technical level, with a slightly lower but still high reliability score. This indicates a well-balanced, rigorous, and informative lecture.

Reliability 9/10