Prime gap

This article uses technical mathematical notation for logarithms. All instances of

log(x)

without a subscript base should be interpreted as a natural logarithm, also commonly written as

ln(x)

or

log e (x)

.

Prime gap frequency distribution for primes up to 1.6 billion. Peaks occur at multiples of 6.^[1]

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)-st and the n-th prime numbers, i.e.

g_{n}=p_{n+1}-p_{n}.\

We have g₁ = 1, g₂ = g₃ = 2, and g₄ = 4. The sequence (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 every prime can be written as

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

^ Ares, Saul; Castro, Mario (February 1, 2006). "Hidden structure in the randomness of the prime number sequence?". Physica A: Statistical Mechanics and Its Applications. 360 (2): 285–296. arXiv:cond-mat/0310148. Bibcode:2006PhyA..360..285A. doi:10.1016/j.physa.2005.06.066. S2CID 16678116.

[1] Ares, Saul; Castro, Mario (February 1, 2006). "Hidden structure in the randomness of the prime number sequence?". Physica A: Statistical Mechanics and Its Applications. 360 (2): 285–296. arXiv:cond-mat/0310148. Bibcode:2006PhyA..360..285A. doi:10.1016/j.physa.2005.06.066. S2CID 16678116.

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