Twin Primes and Prime Triples

A twin prime is a prime number that differs from another prime number by two, like for example, the twin prime pair (41, 43). Twin primes appear despite the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger due to the prime number theorem, which states that the "average gap" between primes less than $n$ is $log(n)$.

The twin prime conjecture, which postulates that there exist infinitely many twin primes, has been one of the great open questions in number theory for many years. In 1849 de Polignac made the more general conjecture that for every natural number $k$, there exist infinitely many prime pairs $p$ and $p′$ such that $p′ − p = 2k$. The special case $k = 1$ is known as the twin prime conjecture.

The Hardy–Littlewood conjecture (named after G. H. Hardy and John Littlewood) is a generalization of the twin prime conjecture. It is concerned with the distribution of prime constellations, including twin primes, in analogy to the prime number theorem.

The largest know twin prime pair found till date is $3756801695685 · 2^{666669} ± 1$. The numbers have $200700$ decimal digits each. There are known to be $808,675,888,577,436$ twin prime pairs below $10^{18}$. The first few twin prime pairs are as follows:

$(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), \dots $

When it comes to prime triples (three consecutive primes, each differing from the previous one by 2), however, it can easily be proved that $3,5,7$ is the only prime triple. The proof follows -

Let us assume, for the sake of argument, that there exists another prime triple apart from 3,5,7.

Also, let $p$, where $p>3$, denote the smallest prime which is a part of that triple.

Then, clearly, this prime triple consists of the three integers $p$, $p+2$ and $p+4$.

Now, by the Division Theorem, we know that any integer $p$ can be expressed in one of the forms $3k$, $3k+1$, or $3k+2$, where $k$ is an integer. Let us consider three distinct cases - 

Case 1: The integer $p$ can be expressed in the form $3k$, where $k$ is an integer.

In this case, the integer $p=3k$ is divisible by 3. 

Case 2: The integer $p$ can be expressed in the form $3k+1$, where $k$ is an integer.

In this case, the integer $p+2=3k+3=3(k+1)$ is divisible by 3. 

Case 3: The integer $p$ can be expressed in the form $3k+2$, where $k$ is an integer.

In this case, the integer $p+4=3k+6=3(k+2)$ is divisible by 3. 

So, we have proved that at least one of the integers $p$, $p+2$, $p+4$ is divisible by 3. But this is obviously a contradiction, since each of the integers $p$, $p+2$, and $p+4$ are supposed to be primes strictly greater than 3. Thus, our assumption must have been incorrect.

Hence, we have proved that the only prime triple is 3,5,7.