Andrica's conjecture

Andrica's conjecture (named after Romanian mathematician Dorin Andrica (es)) is a conjecture regarding the gaps between prime numbers.

denotes the nth prime gap, then Andrica's conjecture can also be rewritten as

Imran Ghory has used data on the largest prime gaps to confirm the conjecture for

[2] Using a more recent table of maximal gaps, the confirmation value can be extended exhaustively to 2×1019 > 264.

Since the Andrica function decreases asymptotically as n increases, a prime gap of ever increasing size is needed to make the difference large as n becomes large.

As a generalization of Andrica's conjecture, the following equation has been considered: where

is the nth prime and x can be any positive number.

The largest possible solution for x is easily seen to occur for n=1, when xmax = 1.

The smallest solution for x is conjectured to be xmin ≈ 0.567148... (sequence A038458 in the OEIS) which occurs for n = 30.