The ternary Goldbach problem

The ternary Goldbach conjecture, or three-primes problem, states that every odd number $n$ greater than $5$ can be written as the sum of three primes. The conjecture, posed in 1742, remained unsolved until now, in spite of great progress in the twentieth century. In 2013 -- following a line of research pioneered and developed by Hardy, Littlewood and Vinogradov, among others -- the author proved the conjecture. In this, as in many other additive problems, what is at issue is really the proper usage of the limited information we possess on the distribution of prime numbers. The problem serves as a test and whetting-stone for techniques in analysis and number theory -- and also as an incentive to think about the relations between existing techniques with greater clarity. We will go over the main ideas of the proof. The basic approach is based on the circle method, the large sieve and exponential sums. For the purposes of this overview, we will not need to work with explicit constants; however, we will discuss what makes certain strategies and procedures not just effective, but efficient, in the sense of leading to good constants. Still, our focus will be on qualitative improvements.Comment: 29 pages. To be submitted to the Proceedings of the ICM 201

On Grosswald's conjecture on primitive roots

Grosswald's conjecture is that $g(p)$, the least primitive root modulo $p$, satisfies $g(p) \leq \sqrt{p} - 2$ for all $p>409$. We make progress towards this conjecture by proving that $g(p) \leq \sqrt{p} -2$ for all $409<p< 2.5\times 10^{15}$ and for all $p>3.67\times 10^{71}$.Comment: 7 page

The ternary Goldbach conjecture is true

The ternary Goldbach conjecture, or three-primes problem, asserts that every odd integer $n$ greater than $5$ is the sum of three primes. The present paper proves this conjecture. Both the ternary Goldbach conjecture and the binary, or strong, Goldbach conjecture had their origin in an exchange of letters between Euler and Goldbach in 1742. We will follow an approach based on the circle method, the large sieve and exponential sums. Some ideas coming from Hardy, Littlewood and Vinogradov are reinterpreted from a modern perspective. While all work here has to be explicit, the focus is on qualitative gains. The improved estimates on exponential sums are proven in the author's papers on major and minor arcs for Goldbach's problem. One of the highlights of the present paper is an optimized large sieve for primes. Its ideas get reapplied to the circle method to give an improved estimate for the minor-arc integral.Comment: 79 pages, 1 figure. Minimal change

Computational sieving applied to some classical number-theoretic problems

Many problems in computational number theory require the application of some sieve. Efficient implementation of these sieves on modern computers has extended our knowledge of these problems considerably. This is illustrated by three classical problems: the Goldbach conjecture, factoring large numbers, and computing the summatory function of the M'{obius function

A SET OF NEW SMARANDACHE FUNCTIONS, SEQUENCES AND CONJECTURES IN NUMBER THEORY

The Smarandache's universe is undoubtedly very fascinating and is halfway between the number theory and the recreational mathematics. Even though sometime this universe has a very simple structure from number theory standpoint, it doesn't cease to be deeply mysterious and interesting