Twin Primes
A twin prime is a prime number \(p\) such that either \(p+2\) or \(p-2\) is also prime. For example, \(\{3,5\}\), \(\{5, 7\}\), and \(\{11,13\}\) are pairs of twin primes. Twin primes are the subject of the twin prime conjecture, which hypothesizes that there are infinitely many pairs of twin primes. This conjecture remains unsettled, but significant progress was made in 2013, when Yitang Zhang proved that there exists an integer \(N < 7 \cdot 10^7\) such that there are infinitely many primes \(p\) for which \(p+N\) is also a prime number. Since then, an online collaborative project (through Polymath) has improved this result to \(N < 246\).
Counting Twin Primes
The twin prime counting function is defined as
\[\pi_2 (n) := \{\text{number of twin primes } \le n\}.\]
One may reformulate the twin prime conjecture using this function: the conjecture states \( \lim_{n\to\infty} \pi_2 (n) = \infty\).
By the prime number theorem, the probability that a number \(n\) is prime should be approximately \(\frac1{\log n}\) for large \(n\). This is because the prime number theorem states that \(\pi(n) \sim \frac{n}{\log n}\), where the prime counting function \(\pi(n)\) is defined to be the number of primes \(\le n\). If the events
\[\{n \text{ is prime}\}\quad \text{ and }\quad \{n+2 \text{ is prime}\}\]
were independent, then one would deduce that the probability \(n\) is a twin prime is \(\frac1{(\log n)^2}\), so \(\pi_2 (n) \sim \frac{n}{(\log n)^2}\).
In fact, this asymptotic approximation is slightly inaccurate. The correct (conjectured) asymptotic approximation is
\[\pi_2 (n) \sim 2C \frac{n}{(\log n)^2}, \]
where
\[C =\prod_{\substack{p\ge 3 \\ p \text{ prime}}} \frac{p(p-2)}{(p-1)^2} \approx 0.6601. \]
This \(C\) is called the twin prime constant. If this asymptotic were shown to hold, then the twin prime conjecture would follow, since \(\frac{n}{(\log n)^2} \to \infty\) as \(n\to\infty\).
Brun's Theorem
Another potential approach to proving the twin prime conjecture is as follows. Consider the sum of the reciprocals of the twin primes:
\[\sum_{p \text{ smaller twin prime}} \left( \frac{1}{p} + \frac{1}{p+2} \right).\]
If this sum diverges, then there must be infinitely many twin primes.
Accordingly, Viggo Brun decided to study this sum, and in 1919, he proved that the sum converges. The value of the sum, known as Brun's constant, is approximately \(1.902\). Because the sum converges, it is not possible to conclude from studying this sum that there are infinitely many twin primes. However, if someone were to prove the irrationality of Brun's constant, that would be a sufficient condition to conclude that there are infinitely many twin primes.