Much of number theory is about trying to understand the interaction between two different structures on the integers (or the natural numbers): the additive structure and the multiplicative structure.

To clarify what I mean, consider the twin prime conjecture, one of the famous open problems in analytic number theory. The conjecture asserts that there are infinitely many twin primes, in other words, primes $p$ such that $p + 2$ is also prime. Being a prime number is a multiplicative property: a natural number $\geq 2$ is defined to be prime if it's only divisible by itself and $1$ . In contrast, adding two to a number is an additive operation. Therefore the twin prime conjecture is asking a question about how these two structures interact with each other.

A substantial chunk of the open problems in number theory are about trying to get a better understanding of the interaction between these two structures. In this notebook, I'll be focusing on one of the most powerful conjectures in this area: the $a b c$ conjecture.

Randomness

Before I talk about any specific problems, I'll take the time to introduce the concept of heuristics based on randomness. These are suggestive probabilistic arguments that "prove" a result under false assumptions that different events are independent when they are really not so.

Let's see how we can "prove" the twin prime conjecture using such methods. Suppose that we pick a large natural number $N$ and we pick a natural number $n$ from the interval $[N, 2 N]$ uniformly at random. What's the chance that $n$ and $n + 2$ form a twin prime pair?

Well, we know by the prime number theorem that there are roughly $\sim N / log N$ primes in this interval, so if we say $E_{1}$ is the event of $n$ being prime and $E_{2}$ is the event of $n + 2$ being prime, we'll have roughly $P (E_{1}) \approx P (E_{2}) \sim 1 / log N$ . We want the probability that both of them are prime, in other words, we want $P (E_{1} \cap E_{2})$ .

So far everything we've done is rigorous. Now we'll introduce the heuristic: we'll assume that the events $E_{1}$ and $E_{2}$ are independent, which amounts to assuming that in this case the additive and multiplicative structures on the integers are "orthogonal" in some sense. Under this assumption, we can deduce

$P (E_{1} \cap E_{2}) = P (E_{1}) \cdot P (E_{2}) \sim \frac{1}{{log}^{2} (N)}$

In other words, the probability of a natural number chosen uniformly at random from $[N, 2 N]$ being the smaller of a pair of twin primes is some constant times $1 / {log}^{2} (N)$ . That tells us that there are roughl...

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