Book Brief · P3 (Prompt → Program → Proof)

Proofs from THE BOOK — Sixth Edition

Martin Aigner & Günter M. Ziegler · Springer · 326 pp

Elegant Proofs Number Theory Geometry Combinatorics

P3 Summary: Prompt → Program → Proof

Prompt & Question

Prompt: Create a concise, self-contained brief of Proofs from THE BOOK (6e) with examples of topics and the aesthetic idea behind “The Book”.

Question: What is “The Book” idea attributed to Erdős, what’s new in the sixth edition, and what range of theorems does the book cover?

Data (Sources)

  • SpringerLink product page (official metadata, ToC with 45 chapters, new material notes).[1]
  • Overview of the Erdős “Book” idea, editions, languages, examples, and Steele Prize note.[2]
  • AMS announcement of the 2018 Leroy P. Steele Prize for Mathematical Exposition awarded for this book.[3]

Logic (How we evaluate)

  1. Use the Springer page for bibliographic facts, pages, ISBNs, ToC, and “what’s new”.
  2. Use the AMS notice and Wikipedia for context: Erdős’s concept and the prize.
  3. When citing particular chapter topics, reference the ToC entries on Springer.

Program (Driver)

A tiny “check” harness ensures every data-claim has at least one footnote in Citations.

// Pseudocode
const claims=[...document.querySelectorAll('[data-claim]')];
for (const c of claims) assert(c.querySelector('sup a[href^="#fn-"]'));
Proof = Reason Why + Check. “Reason Why” summarizes evidence; “Check” verifies structure.

Answer (TL;DR + Argument Map)

Aigner & Ziegler curate elegant, insight-first proofs—the kind Erdős imagined in “The Book”—spanning number theory, geometry, analysis, combinatorics, and graph theory; the sixth edition adds new material (e.g., a chapter on van der Waerden’s permanent conjecture).[1], [2]

  1. The Book idea: Homage to Erdős’s mythical collection of the most beautiful proofs.[2]
  2. Organization: 45 chapters across major fields; multiple proofs per gem to contrast methods.[1]
  3. What’s new in 6e: New chapter on the permanent conjecture, new sections (e.g., Latin squares), new Basel proof.[1]
  4. Recognition: AMS Steele Prize for Mathematical Exposition (2018) awarded for this book.[3]

Reason Why (Evidence)

Springer lists it as the revised and enlarged sixth edition with the new permanent-conjecture chapter, new sections on the asymptotics of Latin squares, and a new Basel-problem proof; the ToC shows 45 chapters.[1]

The “Book” idea—God’s collection of the most beautiful proofs—is widely attributed to Paul Erdős; the book’s title is an homage to that concept and the authors received the AMS Steele Prize (2018) for exposition for this work.[2], [3]

Check (Self-test)

Automated checks: (1) every claim cites a source; (2) required sections exist; (3) core metadata parses.

    Running checks…

    Math Track: Representative Gems in the Sixth Edition

    1. Six proofs of the infinitude of primes. A tour from Euclid-style arguments to topological and analytic twists (chapter listed in ToC).[1]
    2. Quadratic reciprocity. Several clean proofs illuminate the law’s symmetry for odd primes.[1], [4]
    3. Wedderburn’s little theorem. Every finite division ring is a field—an algebraic highlight with short proofs.[1], [5]
    4. Basel problem, new proof. The classic sum of inverse squares gets a fresh treatment in this edition.[1], [6]
    5. Hilbert’s third problem. Decomposing polyhedra—an elegant geometric chapter.[1]
    6. Spectral theorem & Hadamard’s determinant problem. Linear-algebraic vistas with crisp arguments.[1]
    7. New chapter: van der Waerden’s permanent conjecture. A modern jewel added in the sixth edition.[1]
    Takeaway: The book emphasizes short, surprising arguments and multiple proofs, illustrating mathematical beauty via contrast and economy.[1], [2]

    Themes

    1. Beauty and brevity. The curatorial lens is elegance first—proofs selected for insight and simplicity.[1]
    2. Many roads to truth. Multiple independent proofs of the same theorem highlight structure and intuition.[1], [2]
    3. Cross-field unity. Number theory, geometry, analysis, combinatorics, and graph theory sit side by side, reinforcing shared ideas.[1]

    Studies & Context

    • Editions and scope. From the first edition to the sixth (45 chapters), with translations into many languages.[2], [1]
    • Recognition. AMS Steele Prize for Mathematical Exposition awarded to Aigner and Ziegler for this book (2018).[3]

    Glossary (quick reference)

    The Book (Erdős)
    The imagined compendium of the most beautiful proofs; the book’s title is an homage to this idea.[2]
    Quadratic reciprocity
    Symmetry law describing when one prime is a quadratic residue modulo another; a central result in elementary number theory.[4]
    Basel problem
    Evaluation of the series 1 + 1/2² + 1/3² + … = π²/6; featured with a new proof in this edition.[1], [6]
    Wedderburn’s little theorem
    Every finite division ring is a field; appears among the algebraic highlights.[5], [1]

    Book Metadata

    • Title: Proofs from THE BOOK (Sixth Edition).[1]
    • Authors: Martin Aigner; Günter M. Ziegler.[1]
    • Publisher: Springer Berlin, Heidelberg.[1]
    • Publication: eBook 2018-06-14; hardcover 2018-07-06.[1]
    • Pages: 326 (VIII + 326); ISBN (HC): 978-3-662-57264-1; ISBN (eBook): 978-3-662-57265-8.[1]
    • Scope: 45 chapters across number theory, geometry, analysis, combinatorics, graph theory.[1]

    Citations (for this page)

    1. SpringerLink — Proofs from THE BOOK (6th ed.) — metadata, ToC (45 chapters), new material notes.
    2. Wikipedia — background on the title’s Erdős theme; examples; editions and translations.
    3. AMS Notices — 2018 Leroy P. Steele Prizes (Exposition prize to Aigner & Ziegler for this book).
    4. Wikipedia — Quadratic reciprocity (overview).
    5. Wikipedia — Wedderburn’s little theorem (overview).
    6. Wikipedia — Basel problem (overview).

    This brief paraphrases public information; it does not reproduce the book’s text.