| 1, 2, 3, 4, 5 — mathematics begins with counting. Next comes addition, then multiplication. At first glance, they seem quite similar. After all, multiplication is just repeated addition: 7 × 5 is a shorter way to write 5 + 5 + 5 + 5 + 5 + 5 + 5. But if you glimpse inside numbers to see what they are made of, this initial similarity crumbles. Try partitioning any whole number into smaller parts using addition, and you'll have a rich buffet of options. For example, 11 = 5 + 6 = 4 + 7 = 3 + 3 + 3 + 2. (There are 56 ways to split up 11.) As numbers get bigger, the number of partitions steadily grows. But if you instead try to split numbers up using multiplication, a very different picture emerges. There are lots of ways to split up 30 — there's 3 × 10 and 5 × 6 and 2 × 15. But 31 can't be split at all. It is prime. Its only factors are itself and 1. This distinction between addition and multiplication is one of the gentlest mountain passes into the high wilderness of abstract mathematics. The definition of prime numbers involves multiplication. But the primes also form additive patterns with a mysterious texture. Many of these patterns have motivated math's biggest open problems. For instance, mathematicians suspect that there are infinitely many twin primes — primes (multiplicative) that differ by 2 (additive), like 29 and 31 or 41 and 43. But nobody has been able to show this for sure. Similarly, mathematicians think every even number greater than 2 can be written as the sum of two primes, a problem called Goldbach's conjecture. This, too, remains unproved. But numerous other facts are well established. There are infinitely many prime numbers. Mathematicians continue to come up with new proofs of this, even though it's one of the oldest results in math. Primes are also known to get scarcer along the number line. In 1896, Jacques Hadamard and Charles-Jean de la Vallée Poussin independently proved the so-called prime number theorem, which establishes a very good estimate of how rare they get. This theorem is one of the foundational results of analytic number theory, a branch of math that connects the study of integers with smoothly changing functions.
What's New and Noteworthy
At first glance, integers and functions should have little to do with one another. But the connection between them is profound. One of its most tantalizing strands is the Riemann hypothesis, arguably the most important (and intractable) open question in modern math. On its surface, the hypothesis has nothing to do with prime numbers — it's about the behavior of an infinite sum that does not directly involve primes. But if it's true, mathematicians will have a way to account for deviations from the predictions of the prime number theorem. The primes appear to be haphazardly scattered among the integers, but the Riemann hypothesis provides a sort of gnomic key that explains why they appear when they do. In May, James Maynard and Larry Guth proved a new bound on possible exceptions to the hypothesis. (Physicists also have ideas for how to tackle it.) Last year, three of Maynard's students proved a new result about how primes are distributed into different kinds of mathematical buckets. Still other lines of work examine how primes are distributed in shorter intervals. It's long been known that primes form clumps — sometimes they leave big gaps between them, and sometimes small ones. In 2013, Yitang Zhang, then an unknown mathematician, proved that there are an infinite number of primes separated by less than 70 million numbers. This was the first major step toward showing that there are an infinite number of twin primes: 70 million, though a large number, is finite. A few months later, a collaboration including Maynard showed that it's possible to do a bit better: They shrank the gap from 70 million to 600. Equally interesting to mathematicians is the question of how far apart primes can be. (Even if some primes are spaced closely together, other pairs of adjacent primes are far apart.) The average spacing tends toward infinity for large numbers, but mathematicians try to characterize how quickly gaps can grow. The primes create many patterns beyond just how they are distributed. Except for 2, all the primes are odd. This means that some, like 5, leave a remainder of 1 when divided by 4, while others, like 11, leave a remainder of 3. It turns out that these two different kinds of primes have fundamentally different behaviors, a fact called quadratic reciprocity, which was first proved by Carl Gauss in the 19th century. Reciprocity is a basic tool for mathematicians today. For instance, it played a key role in a proof last summer on how circles can be packed together. The notion of being prime, or indivisible, isn't just limited to numbers. Expressions called polynomials, like x⁵ + 3x² + 1, can be prime too. In 2018, two mathematicians showed that almost all polynomials in a particular class are prime. It is not obvious, at first glance, just how special prime numbers are. As you count, it seems like a curiosity that, say, 7 and 11 are indivisible in a way that other numbers aren't. But the simple act of counting creates subtle and complex structures that allow anybody to glimpse the inexorable grandeur of mathematical truth. | 
