Properties of Prime Numbers
Author: Zhiming Ou
Publisher: 3265 Public Way
ISBN:978-1-997036-02-9
Summary
Everybody knows what is a prime number; but even the smartest human does not know all properties of prime numbers: this is just number theory: anybody can ask a question, but 1000 wise men cannot answer.
Nowadays, any AI can list some known properties of prime numbers, such as the fundamental theorem of Arithmetic; Eratosthenes’ sieve; the prime number theorem (about the number of primes <=x); estimation for the gap between consecutive primes; the distribution of prime numbers in arithmetic sequences; and so on.
A famous problem about prime numbers is the Goldbach Conjecture: any large even number can be written as the sum of 2 odd prime numbers—the so-called “1 + 1” problem. Before this, many mathematicians worked out problems like “2 + 2”, “1 + 2”.
In 1937, the soviet-union mathematician Ivan Matveyevivh Vinogradov proved that, every sufficiently large odd integer can be expressed as the sum of three prime numbers. In 2013, Harald Andres Helfgott removed the condition “sufficiently large”.
Vinogradov’s proof was important because it introduced powerful methods in analytic number theory, especially the Hardy–Littlewood circle method and exponential sum techniques, which influenced much later work on primes. I developed new methods for estimating trigonometric sums.
In 1979, A Chinese mathematician Hua, Luo Geng reported his direct method for solving the Goldbach Conjecture in Cambridge University, England. I learned his idea from one of his students, Na, Ji Sheng, around 1988. Professor Na published a paper <On the estimation of a sum> in Ke Xue Tong Bao. I improved Na’s method, and completely solved Goldbach Conjecture. One purpose of this writing is to show the detailed proof for Goldbach Conjecture.
Prime numbers are closely related with the Riemann Zeta-Function. With the expression of the nontrivial zeroes of this function, one can obtain an asymptotic expansion for the distribution of prime numbers.
For completeness, I also include some known results about prime numbers in this book.
Content
Chapter 1 Known facts about Prime Numbers
Prime Test and Factorization
Special Prime numbers
Functions that generate Primes
Distribution of Prime Numbers
Number of Primes in an Arithmetic Sequence
Unsolved Problems about prime numbers
Chapter 2 Congruence
Properties
Residue Systems
Quadratic residues
Divisibility testing functions
Prime testing functions
Chapter 3 Number of Solutions of Diophantine Equations
The generating-function method for linear equations
The trigonometric-sum method
Hardy-Littlewood Circle method
Waring’s Problem
Chapter 4 Hua’s Method for Solving the Goldbach Conjecture
Chapter 5 Analytic methods for studying arithmetic functions
Analytic Continuation
Asymptotic Expansion
Conversion between Sums and Products