Two Compact Incremental Prime Sieves

A prime sieve is an algorithm that finds the primes up to a bound $n$. We say that a prime sieve is incremental, if it can quickly determine if $n+1$ is prime after having found all primes up to $n$. We say a sieve is compact if it uses roughly $\sqrt{n}$ space or less. In this paper we present two new results: (1) We describe the rolling sieve, a practical, incremental prime sieve that takes $O(n\log\log n)$ time and $O(\sqrt{n}\log n)$ bits of space, and (2) We show how to modify the sieve of Atkin and Bernstein (2004) to obtain a sieve that is simultaneously sublinear, compact, and incremental. The second result solves an open problem given by Paul Pritchard in 1994

The Pseudosquares Prime Sieve

We present the pseudosquares prime sieve, which finds all primes up to n

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

The pseudosquares prime sieve

Abstract. We present the pseudosquares prime sieve

Powder Technology

Combining two or more granular or powder ingredients requires a suitable mixing process, which can be either free or random flow with no attraction forces between the particles or interactive or orderly with the presence of large active particles that attract others forming stable clumps. Food systems have very complex properties that make it difficult to standardize the mixing process. In order to achieve an efficient mixture, diffusive and convective mechanisms must be combined, and its success is achieved with a predominance of homogenization over segregation. Powder products are typically used in industry as dispersion in a liquid and should have some properties such as good wettability, water incorporation, flowability, and instantization. To work with powder products, it is necessary to make determinations such as density, particle size, texture, and compaction force, among others. All these physical properties affect and determine the behavior of powdered products during storage, handling, and processing

Explicit Methods in Number Theory

These notes contain extended abstracts on the topic of explicit methods in number theory. The range of topics includes asymptotics for field extensions and class numbers, random matrices and L-functions, rational points on curves and higher-dimensional varieties, and aspects of lattice basis reduction