= Prime gap
{title2=$g_n=p_{n+1}-p_n$}
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 $m\ge2$, the consecutive <integers> $m!+2,\ldots,m!+m$ 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.
Back to article page
