Introduction to number theory lecture 32. Calculation of the Legendre symbol

Introduction to number theory lecture 32. Calculation of the Legendre symbol

by Richard E Borcherds

Formal & Physical Sciences Mathematics PBMathematicsPBHNumber theory
🎙 Richard E Borcherds 👥 82K 📅 March 13, 2022 ⏱ 28 min 👁 4K 📄 lecture 🧭 2026-08-17
Available in: English (current) Français

Keywords

Legendre symbolquadratic residueGauss's lemmaFermat numbersquadratic reciprocity

Summary

This lecture, part of a Berkeley undergraduate number theory course, focuses on calculating the Legendre symbol for various values. The instructor begins by recalling the definition and the result for 2 from the previous lecture. He then computes the Legendre symbol for -2, 3, -3, 5, and 6 using Gauss’s lemma and multiplicativity. For each case, he derives conditions on the prime p modulo certain numbers (e.g., 8, 12, 20, 24). Applications include proving the infinitude of primes in certain arithmetic progressions (e.g., 7 mod 8, 3 mod 8) and a primality test for Fermat numbers. The lecture concludes by hinting at the law of quadratic reciprocity, which will be covered next.

112 words

Critical Evaluation

Value of the Information & Strength of the Argument

The lecture provides a thorough and rigorous derivation of the Legendre symbol for several small integers, using Gauss’s lemma and multiplicativity. The argumentation is clear and step-by-step, with careful handling of cases. The applications, such as proving infinitude of primes in certain residue classes and the Fermat number primality test, demonstrate the utility of the results. The presentation is mathematically sound and well-motivated.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is based on the standard textbook ‘An Introduction to the Theory of Numbers’ by Niven, Zuckerman, and Montgomery. The instructor is a well-known mathematician, and the content is rigorous. The title accurately reflects the content. No external sources are cited beyond the textbook and the course playlist.

127 words

Title / Content Match

The title accurately describes the lecture's focus on calculating the Legendre symbol.

Quality & Reliability

9/10

Lecture by a renowned mathematician, based on a standard textbook, with rigorous proofs and clear derivations. The content is well-structured and mathematically sound.

Key Moments

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

Cited Sources

Concurring Sources

  • An Introduction to the Theory of Numbers (5th edition) — The textbook referenced by the instructor, which covers the same material in detail.

Contribution & Novelties

The lecture provides a clear and systematic method for calculating Legendre symbols using Gauss’s lemma, with applications to prime distribution and Fermat numbers. It sets the stage for the law of quadratic reciprocity.

Pour aller plus loin :

  • Quadratic reciprocity — The law that generalizes the calculations shown, allowing efficient computation of Legendre symbols.
  • Gauss’s lemma (number theory) — The key tool used in the lecture to compute Legendre symbols.
  • Fermat number — The numbers for which a primality test is given, with known properties and open questions.
  • Dirichlet’s theorem — The theorem that guarantees infinitely many primes in arithmetic progressions, which the applications illustrate for specific cases.

108 words

Radar Profile

The radar profile shows high scores in information quantity, quality, and reliability, with a slightly lower but still strong technical level. This indicates a dense, rigorous, and well-sourced lecture suitable for an advanced undergraduate audience.

Reliability 9/10