Author | Ribenboim, Paulo. author |
---|---|
Title | The Book of Prime Number Records [electronic resource] / by Paulo Ribenboim |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1988 |
Connect to | http://dx.doi.org/10.1007/978-1-4684-9938-4 |
Descript | XXIII, 476 p. online resource |
1. How Many Prime Numbers Are There? -- I. Euclidโs Proof -- II. Kummerโs Proof -- III. Pรณlyaโs Proof -- IV. Eulerโs Proof -- V. Thueโs Proof -- VI. Two-and-a-Half Forgotten Proofs -- VII. Washingtonโs Proof -- VIII. Fรผrstenbergโs Proof -- 2. How to Recognize Whether a Natural Number Is a Prime? -- I. The Sieve of Eratosthenes -- II. Some Fundamental Theorems on Congruences -- A. Fermatโs Little Theorem and Primitive Roots Modulo a Prime -- B. The Theorem of Wilson -- C. The Properties of Giuga, Wolstenholme and Mann & Shanks -- D. The Power of a Prime Dividing a Factorial -- E. The Chinese Remainder Theorem -- F. Eulerโs Function -- G. Sequences of Binomials 31 -- H. Quadratic Residues -- III. Classical Primality Tests Based on Congruences -- IV. Lucas Sequences -- V. Classical Primality Tests Based on Lucas Sequences -- VI. Fermat Numbers -- VII. Mersenne Numbers -- VIII. Pseudoprimes -- Carmichael Numbers -- X. Lucas Pseudoprimes -- XI. Last Section on Primality Testing and Factorization! -- 3. Are There Functions Defining Prime Numbers? -- I. Functions Satisfying Condition (a) -- II. Functions Satisfying Condition (b) -- III. Functions Satisfying Condition (c) -- 4. How Are the Prime Numbers Distributed? -- I. The Growth of ?(x) -- II. The nth Prime and Gaps -- III. Twin Primes -- IV. Primes in Arithmetic Progression -- V. Primes in Special Sequences -- VI. Goldbachโs Famous Conjecture -- VII. The Waring-Goldbach Problem -- VIII. The Distribution of Pseudoprimes and of Carmichael Numbers -- 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 ร 2โ ยฑ 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. Some Probabilistic Estimates -- IV. The Density of the Set of Regular Primes -- Conclusion -- Dear Reader: -- Citations for Some Possible Prizes for Work on the Prime Number Theorem -- A. General References -- B. Specific References -- 1 -- 2 -- 3 -- 4 -- 5 -- 6 -- Conclusion -- Primes up to 10,000 -- Index of Names -- Gallimawfries