Prime Number Gap Conjectures

A prior post discussed gaps between successive prime numbers and included a spreadsheet for exploring these gaps for all primes under 10,000. In this modest set of primes, the largest gap we see is 36. Seeing these gaps grow raises the question of how large the gap between successive primes can be. The gap may grow without limit. As of August 2018, the largest known prime gap has length 1550, found by Bertil Nyman. This gap occurs between the prime 18,361,375,334,787,046,697 and the next prime.

Several theories have been proposed to explain the gaps between prime numbers. One is Andrica’s Conjecture, which states for all primes:

This difference is given in the linked spreadsheet for all primes under 10,000. While these differences vary empirically, they appear to max out at approximately 0.670873.

Another popular theory is Legendre’s Conjecture, which states that for all n, a prime number exists between:

The linked spreadsheet does not test Legendre’s conjecture, but readers are invited to modify it in their exploration of prime numbers.

Math comedian Matt Parker has a good video on prime number gaps.

A Mathematica program to generate the first 1000 prime gaps is:

Table[Prime[n+1]-Prime[n],{n,1000}]