Prime gap

A prime gap is the difference between two successive prime numbers. The ${\displaystyle n}$-th prime gap, denoted ${\displaystyle g_{n}}$ or ${\displaystyle g(p_{n})}$ is the difference between the ${\displaystyle (n+1)}$th and the ${\displaystyle n}$-th prime numbers, i.e.,

${\displaystyle g_{n}=p_{n+1}-p_{n}}$

For example, since the first few primes are 2, 3, 5, 7, 11..., we have ${\displaystyle g_{1}=1}$, ${\displaystyle g_{2}=g_{3}=2}$, ${\displaystyle g_{4}=4}$. The sequence ${\displaystyle g_{n}}$ of prime gaps has been extensively studied; however, many questions and conjectures remain unanswered.

The first 60 prime gaps are:

1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, ... (sequence A001223 in the OEIS).

By the definition of ${\displaystyle g_{n}}$ every prime can be written as

${\displaystyle p_{n+1}=2+\sum _{i=1}^{n}g_{i}.}$