According to Wikipedia "Prime gap", https://en.wikipedia.org/wiki/Prime_gap (16-Aug-2020): "A prime gap is the difference between two successive prime
numbers. The n-th prime gap, denoted g_{n} or g(p_{n}) is the difference between
the (n+1)-th and the n-th prime numbers, i.e. g_{n} = p_{n+1} - p_{n} . We have
g_{1} = 1, g_{2} = g_{3} = 2, and g_{4} = 4."

It can be proven that there are arbitrary large gaps, usually by showing that
"in the sequence n!+2, n!+3, ..., n!+n the first term is divisible by 2, the
second term is divisible by 3, and so on. Thus, this is a sequence of n-1
consecutive composite integers, and it must belong to a gap between primes
having length at least n.", see prmgap.

Instead of using the factorial of n (see df-fac), any function can be chosen
for which f(n) is not relatively prime to the integers 2, 3, ..., n. For
example, the least common multiple of the integers 2, 3, ..., n, see
prmgaplcm, or the primorial n# (the product of all prime numbers less than
or equal to n), see prmgapprmo, are such functions, which provide smaller
values than the factorial function (see lcmflefac and prmolefac resp.
prmolelcmf): "For instance, the first prime gap of size larger than 14
occurs between the primes 523 and 541, while 15! is the vastly larger number
1307674368000." But the least common multiple of the integers 2, 3, ..., 15 is
360360, and 15# is 30030 (p_{3248} = 30029 and P_{3249} = 30047, so g_{3248} = 18).