A prime gap is the difference between two successive prime numbers. The n-th prime gap, denoted gn or g(pn) is the difference between the (n + 1)-th and the
n-th prime numbers, i.e.
g
n
=
p
n
+
1
−
p
n
.
{\displaystyle g_{n}=p_{n+1}-p_{n}.\ }
We have g1 = 1, g2 = g3 = 2, and g4 = 4. The sequence (gn) 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 gn 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}.}
View More On Wikipedia.org
