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. It may be possible the gap grows 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.

There are many theories proposed about 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.