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The Goldbach Conjecture

Yuan Wang
World Scientific
Publication Date: 
Number of Pages: 
Series in Pure Mathematics 4
[Reviewed by
David Farmer
, on

This book is a collection of some of the main papers that have contributed to progress on the Goldbach Conjecture. Recall that the Goldbach Conjecture is the assertion that every even integer n ≥ 4 can be written as the sum of two primes. For example, 100 = 3 + 97. A little experimentation reveals that usually there are many ways to write an even integer as the sum of primes, and there has never been any serious doubt that the Goldbach Conjecture is true.

What does it mean to make progress on such a conjecture? For most of the papers in this collection, "progress" involves proving a weaker version of the conjecture, possibly under some additional hypotheses. For example, the Goldbach Conjecture is true, then every odd integer n ≥ 7 is the sum of three primes. Proof: n - 3 is an even integer greater than or equal to 4, so by Goldbach n - 3 = p + q with p and q prime, so n = 3 + p + q. This is called "the ternary Goldbach Conjecture."

Mathematicians have not quite proven the ternary Goldbach Conjecture, but we are absurdly close. We know that all sufficiently large odd integers are the sum of three primes, and the conjecture has been checked up to (at least) 1017. Unfortunately, in this case "sufficiently large" means larger than 1043000, so there is a gap which computers cannot span. (Computers can span the gap if one assumes the Generalized Riemann Hypothesis, but unconditionally there is still a gap.) This book includes a half-dozen papers leading to these results.

Another example of a weaker version of the Goldbach Conjecture is to prove that every even integer is the sum of two "almost primes." That is, numbers having an explicitly bounded number of prime factors. For example, a 4-almost prime is an integer that is the product of at most 4 prime factors. The current record, which is as close as possible without actually proving Goldbach, is that all sufficiently large even integers can be written as the sum of a prime and a 2-almost prime. Papers leading to that result are contained in this volume. Don't hold your breath waiting for the next improvement.

One feature that is clear from the above discussion is that these results tend to be of the form "for all sufficiently large...". The reason for this restriction is that the proofs involve techniques from analytic number theory, and what one proves is a formula (for example) for the number of ways that an odd integer can be written as the sum of three primes. The formula usually involves a main term and an error term, where (hopefully) the main term is growing faster than the error term. So, if one goes out far enough then the main term is larger than the error term, hence the "sufficiently large." These error terms often come from basic results in analytic number theory, such as the large sieve or the Bombieri-Vinogradov inequality, so the book also includes papers on those topics.

In addition to the collected papers, the book contains a useful introduction that summarizes the main ideas in the collection. It would have been more useful if a brief summary of every paper was included. The introduction would also be improved if it was edited by a native English speaker. In most cases the meaning is clear, but the awkwardness detracts from an otherwise nice survey.

Who should buy this book? The book certainly is not suitable for someone who wants to learn about the subject from scratch. Original sources are rarely the best way for a student to learn mathematics, and this topic is no exception. But if you already have some familiarity with this material, and you like the idea of browsing through some of the original sources, then you will like having this book on your shelf.

David Farmer is Director of Web Programming at the American Institute of Mathemtics Research Conference Center.

* Representation of an Odd Number as the Sum of Three Primes:
* A New Proof of the Goldbach-Vinogradov Theorem (J V Linnik)
* A New Proof on the Three Primes Theorem (C B Pan)
* An Elementary Method in Prime Number Theory (R C Vaughan)
* A Complete Vinogradov 3-Primes Theorem under the Riemann Hypothesis (J M Deshouillers et al.)
* Representation of an Even Number as the Sum of Two Almost Primes (Elementary Approach):
* New Improvements in the Method of the Sieve of Eratosthenes (A A Buchstab)
* On Prime Divisors of Polynomials (P Kuhn)
* On an Elementary Method in the Theory of Primes (A Selberg)
* Lectures on Sieves (A Selberg)
* Representation of an Even Number as the Sum of a Prime and an Almost Prime:
* On the Representation of large Integer as a Sum of a Prime and an Almost Prime (Y Wang)
* The Density Hypothesis for Dirichlet L-Series (A I Vinogradov)
* On the Large Sieve (E Bombieri)
* and other articles