Andrica's conjecture

Andrica's conjecture is a conjecture regarding the gaps between prime numbers.
The conjecture states that the inequality
holds for all, where is the -th prime number. If denotes the -th prime gap, then Andrica's conjecture can also be rewritten as

Empirical evidence

Imran Ghory has used data on the largest prime gaps to confirm the conjecture for up to. Using a more recent table of maximal gaps, the confirmation value can be extended exhaustively to > 2⁶⁴.
The discrete function is plotted in the figures opposite. The high-water marks for occur for n = 1, 2, and 4, with A₄ ≈ 0.670873..., with no larger value among the first 10⁵ primes. 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. It therefore seems highly likely the conjecture is true, although this has not yet been proven.
The best proven bound on gap sizes is . Thus, using an inequality of, the conjecture is verified for up to 1.099532599291×10¹².

Generalizations

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 x_max = 1. The smallest solution for x is conjectured to be x_min ≈ 0.567148... which occurs for n = 30.
This conjecture has also been stated as an inequality, the generalized Andrica conjecture: