Prime Numbers: The Holy Grail Of Mathematics: A brief introduction to prime numbers
- Length: 390 pages
- Edition: 1
- Language: English
- Publication Date: 2021-10-09
- ISBN-10: B09J45FDQ7
- Sales Rank: #1234129 (See Top 100 Books)
It is undeniable how prime numbers are one of the most beautiful and fascinating topics in mathematics. But what are prime numbers? Are they only numbers that are divisible by 1 and themselves, or do they have another interesting hidden face?Throughout history, the mystery of prime numbers has challenged the greatest minds in mathematics starting from Euclid of Alexandria to Fermat, Euler, Gauss, and Erdős,… who attempted to solve the puzzling problem of primes. The achievements they realized and the secrets they revealed can only assert how deep the concept of prime numbers is. Starting from how prime numbers exist in nature, and how they are of great use in modern cryptography on which our daily life completely depends, the author travels in the holy kingdom of primes diving into some conjectures involving those special numbers. From the Riemann Hypothesis and the well-known zeta function, he explains how a note in the margin turned to be Fermat’s Last Theorem, one of the most important problems in the history of mathematics. From Mersenne Primes, he gets to the twin primes, those shining little stars in the blue sky of primes. And from Euclid’s proof of the infinite number of primes he gets to a hidden pattern in the distribution of primes discovered by Stanisław Ulam and called the Ulam Spiral. After this little trip, you will know, dear reader, why prime numbers deserve to be called “the holy grail of mathematics”.
List of figures A note from the author Acknowledgments Introduction Chapter 1 : Primes in nature Chapter 2 : Primes and Aliens Chapter 3 : Primes and cryptography Chapter 4 : How many primes are there ? Chapter 5 : The fundamental Theorem of Arithmetic Chapter 6 : Sieving for Primes Sieve of Eratosthenes Sieve of Euler Chapter 7 : The prime-counting function π Chapter 8 : Euler’s Totient function Some properties of φ (n) Perfect totient numbers Carmichael’s totient function conjecture Lehmer’s Totient Problem Chapter 9 : Mersenne Primes The connection between Sophie Germain primes and Mersenne primes Hunting for Mersenne primes How to find Mersenne primes Interesting facts about Mersenne primes Perfect numbers Chapter 10 : Pierre de Fermat Fermat’s theorem on the sum of two squares Fermat’s Last Theorem Generalization of Fermat’s Last Theorem Fermat-Catalan Conjecture Beal’s Conjecture Fermat’s Little Theorem Fermat Quotient Wieferich primes Wieferich numbers Fermat Numbers Applications of Fermat Numbers Generalized Fermat Numbers Chapter 11 : The Riemann Hypothesis Riemann’s work in number theory Chapter 12 : Primes in Arithmetic Progressions Chapter 13 : Dirichlet’s Theorem Dickson’s conjecture: A Generalization of Dirichlet’s Theorem Hypothesis H: A Generalization of Dickson’s conjecture Chapter 14 : Formulas for Primes Mill s’ Theorem Euler’s quadratic Chapter 15 : The Goldbach Conjecture Goldbach Partition and Goldbach Numbers Related problems Goldbach Conjecture in literature Chapter 16 : Bertrand’s Postulate Related problems Legendre’s conjecture Brocard’s conjecture Andrica’s conjecture Oppermann’s conjecture Chapter 17 : Wilson’s Theorem Generalized Wilson’s Theorem Wilson Primes Generalized Wilson Primes Chapter 18 : The Twin Prime Conjecture Generalization De Polignac’s Conjecture Chapter 19 : The Ulam Spiral Variants: Klauber Triangle References Credits and references of figures and images Appendix I: List of the first 1000 prime numbers Index
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