Publications

I'm writing an "open book", related to the Goldbach conjecture. One of the tests already performed is to check that every integer z <= n (n very very large) can be written as z = x + y, where x, y have the form INT(a*k^b) with k integer >=0, a, b > 0. The result seems valid for a large class of b>1 and even for b as large as b=2, provided that a is small enough. This (if proven true for all integers) is stronger than Goldbach's conjecture. When a=2, the result seems to be true when b<=1.35, but not when b>=1.36. This is also stronger than Goldbach. Note that Goldbach's conjecture is equivalent to "every integer z>=0 can be written as z = x + y, where x=(p-1)/2, y=(q-1)/2, p,q prime".

URL: http://datashaping.com/goldbach.shtml

Open Book: Generalized Goldbach Conjecture and Integer Coverages

This book is available for free and can be downloaded from this web page. We expect the book to be completed by December 2008. In the meanwhile, you are welcome to check our progress, add ideas and suggestions (in LaTeX format) and distribute or use this resource for your own needs. It is quite possibly the first time that a mathematical book with state-of-the-art research results about fundamental - yet very deep number theoretical problems - can be accessed, and even modified, while it is in the process of being written.

The book, essentially written by Dr. Vincent Granville, is intended to a wide audience, so we want to use a terminology that many people are familiar with. So, while we occasionally make reference to facts that are very difficult to prove (e.g. about the distribution of sums of squares or prime numbers), our approach is essentially elementary, and based on combinatorics and probability theory more so than on properties of integers (congruences) or classical analytical number theory. We want this book to show to many people the beauty of mathematics and the incredible complexity of fundamental problems, such as every even integer greater than 5 can be expressed as the sum of two primes.

Whether we actually prove the Goldbach conjecture is left for the reader to decide. We make no claims that we have a complete proof though. However, we build the framework for a new theoretical approach to handle the Goldbach conjecture, as well as considerably more general and beautiful integer coverage results. A particular case involves the Waring conjecture.

We've just started to write the book in December 2007. Since this is a self-funded, private project, We might very well publish and distribute the book at our expense when it is completed. The good thing is that we can share our progress with you without fear of being copied. We are not looking for glory, but rather to promote mathematics to the general public, including to undergraduate or even high school students throughout the world, yet undecided about career choices and lacking exposure to fantastic mathematical theories formulated in simple, plain English words.

Download the book in pdf format.

To check our progress, please bookmark this URL. And remember, you are welcome to contribute with original mathematical ideas.

Vincent Granville, Ph.D.
Data Shaping Solutions
Seattle, December 2007