Twin primes

Definition

The term twin primes is used for a pair of odd prime numbers that differ by two. In other words, primes $p,p+2$ are termed twin primes.

The twin primes conjecture states that there are infinitely many twin primes.

Relation with other properties

Related properties for pairs of primes

• Cousin primes are two primes that differ by $4$. Note that for $p>3$, if both $p$ and $p+4$ are prime, $p+2$ is not prime.
• Sexy primes are two primes that differ by $6$ (and with no prime in between).

Related properties for more than two primes

• Prime quadruplet is a collection of four primes $p,p+2,p+6,p+8$.
• Prime constellation is a sequence of consecutive primes $p_{1}<p_{2}<\dots <p_{k}$ for which the difference between the first and last prime is the least possible based on considerations of modular arithmetic relative to smaller primes.