Verification of the Firoozbakht conjecture for primes up to four quintillion

If $p_k$ is the k-th prime, the Firoozbakht conjecture states that the sequence $(p_k)^{1/k}$ is strictly decreasing. We use the table of first-occurrence prime gaps in combination with known bounds for the prime-counting function to verify the Firoozbakht conjecture for primes up to four quintillion $(4\times10^{18})$.Comment: 6 pages, 1 tabl

Refined Goldbach conjectures with primes in progressions

We formulate some refinements of Goldbach's conjectures based on heuristic arguments and numerical data. For instance, any even number greater than 4 is conjectured to be a sum of two primes with one prime being 3 mod 4. In general, for fixed $m$ and $a, b$ coprime to $m$, any positive even $n \equiv a + b \bmod m$ outside of a finite exceptional set is expected to be a sum of two primes $p$ and $q$ with $p \equiv a \bmod m$, $q \equiv b \bmod m$. We make conjectures about the growth of these exceptional sets.Comment: 10 page

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