In the first, a Monthly note, we use Van der Waerden's Theorem and Fermat's Theorem on four squares in an arithmetic progression to give a crazy proof that there are infinitely many primes. In the second we first survey the many proofs, and then develop a dynamical systems approach. The third is survey of the other two for the LMS Newsletter.

The infamous twin prime conjecture states that there are infinitely
many pairs of distinct primes which differ by 2. In April
2013, Yitang Zhang proved the existence of a finite bound \( B \) such that there
are infinitely many pairs of distinct primes which differ by no more than \( B \).
Zhang even showed that one can take B = 70000000.
In November 2013, inspired by Zhang's extraordinary breakthrough, James
Maynard dramatically slashed this bound to 600, by a substantially easier
method. Both Maynard and Terry Tao, who had independently developed the
same idea, were able to extend their proofs to show that for any given integer
\( m \geq 1\) there exists a bound \( B_m\) such that there are infinitely many intervals
of length \( B_m\) containing at least \( m\) distinct primes. We prove this
herein, even showing that one can take \( B_m=e^{8m+5}\).
If Zhang's method is combined with the Maynard-Tao setup, then it appears
that the bound can be further reduced to 246.
This article introduces these results, and explain the arguments
that allow them to prove their spectacular results. The second half
develops a proof of Zhang's main novel contribution, an estimate for
primes in relatively short arithmetic progressions.

We explain the Bhargava composition algorithm for binary quadratic forms, in historical context

As long as people have studied mathematics, they have wanted to know how many primes there are. Getting precise answers is a notoriously difficult problem, and the first suitable technique, due to Riemann, inspired an enormous amount of great mathematics, the techniques and insights permeating many different fields. In this article we will review some of the best techniques for counting primes, centering our discussion around Riemann's seminal paper. We will go on to discuss its limitations, and then recent efforts to replace Riemann's theory with one that is significantly simpler.

An introduction to analytic number theory for Gowers' interesting mathematical writing project

We develop various consequences of the Green-Tao theorem, for example showing that there exist polynomials of any given degree whose first \( k\) values are prime, and proving that there are magic squares of primes of arbitrary size.

This is a survey of what was then known about "prime number races". Subsequently two of my students from that time, Youness Lamzouri and Daniel Fiorilli, have proved several extraordinary results (some with collaborators) greatly extending this theory.

The number of odd entries in a row of Pascal's triangle is always a power of 2. These are either equally split between 1 and 3 mod 4, or are all 1 mod 4. Similarly, for every odd \( a\), the number of entries in a given row of Pascal's trinagle that are \( \equiv a \pmod 8\) is either 0 or a power of 2. We develop a theory to explain this.