Prime gap

A prime gap is the difference between two successive prime numbers. The n {\displaystyle n} -th prime gap, denoted g n {\displaystyle g_{n}} or g ( p n ) {\displaystyle g(p_{n})} is the difference between the ( n + 1 ) {\displaystyle (n+1)} th and the n {\displaystyle n} -th prime numbers, i.e., g n = p n + 1 − p n {\displaystyle g_{n}=p_{n+1}-p_{n}} For example, since the first few primes are 2, 3, 5, 7, 11..., we have g 1 = 1 {\displaystyle g_{1}=1} , g 2 = g 3 = 2 {\displaystyle g_{2}=g_{3}=2} , g 4 = 4 {\displaystyle g_{4}=4} . The sequence g n {\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 g n {\displaystyle g_{n}} every prime can be written as p n + 1 = 2 + ∑ i = 1 n g i . {\displaystyle p_{n+1}=2+\sum _{i=1}^{n}g_{i}.}