 Prime gap

The prime gap $g_n$ is the difference $p_{n+1}-p_n$ between consecutive prime numbers. For the primes $2,3,5,7,11$, the first gaps are $1,2,2,4$. Every gap after the first is even because both endpoint primes are odd.

Prime gaps are unbounded: for any $m\ge 2$, the consecutive integers $m!+2,\dots ,m!+m$ are all composite, since each is divisible by its corresponding summand. Thus there are arbitrarily long prime-free intervals. This elementary argument proves existence without determining how large gaps typically are near a specified bound.

A prime gap is the difference between two successive prime numbers. For example, if \( p_n \) is the \( n \)-th prime number, then the prime gap \( g_n \) between the \( n \)-th and the \( (n+1) \)-th prime can be expressed as: \[ g_n = p_{n+1} - p_n \] Prime gaps can vary significantly in size.