---
tag: 0FER36N8SB
kind: conjecture
name: Twin prime conjecture
msc: [11N05]
sources:
  - url: https://en.wikipedia.org/wiki/Twin_prime
    locator: Twin prime conjecture
---

## Statement

There are infinitely many [[prime numbers|0SN0MAEKYX]] $p$ such that $p + 2$ is also prime.

## Remarks

A pair of primes $(p, p + 2)$ is a pair of twin primes. The first pairs are $(3, 5)$, $(5, 7)$, $(11, 13)$, $(17, 19)$, $(29, 31)$.
The conjecture is the case $k = 1$ of the conjecture of de Polignac: for each positive integer $k$, there are infinitely many primes $p$ such that $p + 2k$ is prime.
The conjecture is open.

## History

De Polignac stated the general conjecture in 1849.
In 2013, Yitang Zhang proved that some gap $N < 7 \cdot 10^7$ occurs between infinitely many pairs of primes. Later work by Maynard and the Polymath project reduced the bound to $246$.

## Explanations

### L1 human

Twin primes are primes with only one number between them, like $41$ and $43$.
The conjecture says that such pairs never stop, even though primes become rarer.