Theory of numbers: Congruences: Euler's theorem

Item: Theory of numbers: Congruences: Euler's theorem
Rating: 5
Author: PostSingularity Institute

by Richard E Borcherds

Formal & Physical Sciences Mathematics PBMathematics PBHNumber theory

🎙 Richard E Borcherds 👥 82K 📅 January 31, 2021 ⏱ 26 min 👁 4K 📄 lecture 🧭 2026-08-17

Available in: English (current) Français

Keywords

Euler's theoremtotient functiongroup theoryLagrange's theoremcyclotomic polynomials

Summary

This lecture is part of an online undergraduate course on the theory of numbers. The speaker, Richard E Borcherds, introduces Euler’s theorem, a generalization of Fermat’s theorem to non-prime moduli. He begins by recalling Fermat’s theorem and its two forms, then states Euler’s theorem: a^φ(m) ≡ 1 (mod m) for a coprime to m, where φ is Euler’s totient function. He explains the proof using group theory, showing that the non-zero residues modulo m form a group under multiplication, and that the order of any element divides the order of the group (Lagrange’s theorem). He illustrates the concept with an example for m=13, using cosets to demonstrate the divisibility. He then notes that Euler’s theorem is not always optimal, giving examples where a smaller exponent works. As applications, he solves a recreational problem to find the last digit of 7^(7^(7^7)), and proves that there are infinitely many primes congruent to 1 modulo 10 (and more generally modulo any prime) using cyclotomic polynomials. He concludes with an exercise for the audience.

170 words

Critical Evaluation

Value of the Information & Strength of the Argument

The lecture provides a clear and rigorous exposition of Euler’s theorem, building on group theory concepts. The argumentation is solid, with each step justified and examples provided to illustrate abstract ideas. The speaker also discusses the limitations of the theorem, enhancing the depth of the content. The applications chosen are interesting and demonstrate the power of the theorem, while also highlighting its non-optimality in certain cases.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is mathematically rigorous, with all claims proven or referenced to standard results. The speaker is a well-known mathematician, adding to the credibility. The title accurately reflects the content, which is focused on Euler’s theorem. No external sources are cited, but the lecture is self-contained and relies on well-established mathematical knowledge. The description provides a link to the full course playlist, which is a useful resource for further study.

151 words

Title / Content Match

The title accurately reflects the content, which focuses on Euler's theorem and its applications.

Quality & Reliability

9/10

Lecture by a renowned mathematician, rigorous proofs, clear explanations, and no unsubstantiated claims.

Key Moments

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

Cited Sources

Theory of Numbers Course Playlist — The lecture is part of this online course; the playlist contains all lectures.

Concurring Sources

Euler's theorem — Standard mathematical reference confirming the statement and proof of Euler's theorem.
Lagrange's theorem — The group theory result used in the proof.

Contribution & Novelties

The lecture provides a clear and rigorous proof of Euler’s theorem using group theory, emphasizing the role of Lagrange’s theorem. It also highlights the non-optimality of Euler’s theorem and demonstrates applications to recreational problems and prime distribution. The use of cyclotomic polynomials to prove infinitude of primes in certain arithmetic progressions is a nice touch.

Pour aller plus loin :

Euler’s theorem — Wikipedia article providing background and alternative proofs.
Lagrange’s theorem (group theory) — The theorem that the order of a subgroup divides the order of the group.
Cyclotomic polynomial — The polynomials used in the proof of infinitely many primes congruent to 1 mod p.
Dirichlet’s theorem on arithmetic progressions — A stronger result that there are infinitely many primes in any arithmetic progression with coprime first term and difference.

132 words

Radar Profile

The radar profile shows high scores in quality and reliability, with slightly lower but still strong scores in quantity and technical level. This indicates a lecture that is both rigorous and informative, though it assumes some prior knowledge of group theory.

Reliability 10/10