The New Book of Prime Number Records

This text originated as a lecture delivered November 20, 1984, at Queen's University, in the undergraduate colloquium senes. In another colloquium lecture, my colleague Morris Orzech, who had consulted the latest edition of the Guinness Book of Records, reminded me very gently that the most "innumerate" people of the world are of a certain trible in Mato Grosso, Brazil. They do not even have a word to express the number "two" or the concept of plurality. "Yes, Morris, I'm from Brazil, but my book will contain numbers different from ·one.''' He added that the most boring 800-page book is by two Japanese mathematicians (whom I'll not name) and consists of about 16 million decimal digits of the number Te. "I assure you, Morris, that in spite of the beauty of the appar ent randomness of the decimal digits of Te, I'll be sure that my text will include also some words." And then I proceeded putting together the magic combina tion of words and numbers, which became The Book of Prime Number Records. If you have seen it, only extreme curiosity could impel you to have this one in your hands. The New Book of Prime Number Records differs little from its predecessor in the general planning. But it contains new sections and updated records.

Autorentext
Paulo Ribenboim ist emeritierter Professor der kanadischen Queen's University, Fellow der Royal Society of Canada und Träger des George Pólya-Preises der Mathematical Association of America. Er ist Autor von 13 Büchern und über 150 Forschungsartikeln.

Klappentext
The Guinness Book made records immensely popular. This book is devoted, at first glance, to present records concerning prime numbers. But it is much more. It explores the interface between computations and the theory of prime numbers. The book contains an up-to-date historical presentation of the main problems about prime numbers, as well as many fascinating topics, including primality testing. It is written in a language without secrets, and thoroughly accessible to everyone. The new edition has been significantly improved due to a smoother presentation, many new topics and updated records.

Zusammenfassung
"Yes, Morris, I'm from Brazil, but my book will contain numbers different from *one.''' He added that the most boring 800-page book is by two Japanese mathematicians (whom I'll not name) and consists of about 16 million decimal digits of the number Te.

Inhalt
1 How Many Prime Numbers Are There?.- I. Euclid's Proof.- II. Goldbach Did It Too!.- III. Euler's Proof.- IV. Thue's Proof.- V. Three Forgotten Proofs.- A. Perott's Proof.- B. Auric's Proof.- C. Métrod's Proof.- VI. Washington's Proof.- VII. Fürstenberg's Proof.- VIII. Euclidean Sequences.- IX. Generation of Infinite Sequences of Pairwise Relatively Prime Integers.- 2 How to Recognize Whether a Natural Number Is a Prime.- I. The Sieve of Eratosthenes.- II. Some Fundamental Theorems on Congruences.- A. Fermat's Little Theorem and Primitive Roots Modulo a Prime.- B. The Theorem of Wilson.- C. The Properties of Giuga, Wolstenholme, and Mann and Shanks.- D. The Power of a Prime Dividing a Factorial.- E. The Chinese Remainder Theorem.- F. Euler's Function.- G. Sequences of Binomials.- H. Quadratic Residues.- III. Classical Primality Tests Based on Congruences.- IV. Lucas Sequences.- V. Primality Tests Based on Lucas Sequences.- VI. Fermat Numbers.- VII. Mersenne Numbers.- VIII. Pseudoprimes.- A. Pseudoprimes in Base 2 (psp).- B. Pseudoprimes in Base a (psp(a)).- C. Euler Pseudoprimes in Base a (epsp(a)).- D. Strong Pseudoprimes in Base a (spsp(a)).- E. Somer Pseudoprimes.- IX. Carmichael Numbers.- X. Lucas Pseudoprimes.- A. Fibonacci Pseudoprimes.- B. Lucas Pseudoprimes (lpsp(P, Q)).- C. Euler-Lucas Pseudoprimes (elpsp(P, Q)) and Strong Lucas Pseudoprimes (slpsp(P, Q)).- D. Somer-Lucas Pseudoprimes.- E. Carmichael-Lucas Numbers.- XL Primality Testing and Large Primes.- A. The Cost of Testing.- B. More Primality Tests.- C. Primality Certification.- D. Fast Generation of Large Primes.- E. Titanic Primes.- F. Curious Primes.- XII. Factorization and Public Key Cryptography.- A. Factorization of Large Composite Integers.- B. Public Key Cryptography.- 3 Are There Functions Defining Prime Numbers?.- I. Functions Satisfying Condition (a).- II. Functions Satisfying Condition (b).- III. Functions Satisfying Condition (c).- IV. Prime-Producing Polynomials.- A. Surveying the Problems.- B. Polynomials with Many Initial Prime Absolute Values.- C. The Prime-Producing Polynomials Races.- D. Primes of the Form m2 + 1.- 4 How Are the Prime Numbers Distributed?.- I. The Growth of ?(x).- A. History Unfolding.- B. Sums Involving the Möbius Function.- C. Tables of Primes.- D. The Exact Value of ?(x) and Comparison with x/(log x), Li(x), and R(x).- E. The Nontrivial Zeros of ?(s).- F. Zero-Free Regions for ?(s) and the Error Term in the Prime Number Theorem.- G. The Growth of ?(s).- H. Some Properties of ?(x).- II. The n th Prime and Gaps.- A. The n th Prime.- B. Gaps Between Primes.- Interlude.- III. Twin Primes.- Addendum on k-Tuples of Primes.- IV. Primes in Arithmetic Progression.- A. There Are Infinitely Many!.- B. The Smallest Prime in an Arithmetic Progression.- C. Strings of Primes in Arithmetic Progression.- V. Primes in Special Sequences.- VI. Goldbach's Famous Conjecture.- VII. The Waring-Goldbach Problem.- A. Waring's Problem.- B. The Waring-Goldbach Problem.- VIII. The Distribution of Pseudoprimes, Carmichael Numbers, and Values of Euler's Function.- A. Distribution of Pseudoprimes.- B. Distribution of Carmichael Numbers.- C. Distribution of Lucas Pseudoprimes.- D. Distribution of Elliptic Pseudoprimes.- E. Distribution of Values of Euler's Function.- 5 Which Special Kinds of Primes Have Been Considered?.- I. Regular Primes.- II. Sophie Germain Primes.- III. Wieferich Primes.- IV. Wilson Primes.- V. Repunits and Similar Numbers.- VI. Primes with Given Initial and Final Digits.- VII. Numbers k×2n±1.- VIII. Primes and Second-Order Linear Recurrence Sequences.- IX. The NSW Primes.- 6 Heuristic and Probabilistic Results about Prime Numbers.- I. Prime Values of Linear Polynomials.- II. Prime Values of Polynomials of Arbitrary Degree.- III. Polynomials with Many Successive Composite Values.- IV. Partitio Numerorum.- V. Some Probabilistic Estimates.- A. Distribution of Mersenne Primes.- B. The log log Philosophy.- VI. The Density of the Set of Regular Primes.- Conclusion.- The Pages That Couldn't Wait.- Primes up to 10,000.- Index of Tables.- Index of Names.

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