Andrica's conjecture

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

The conjecture states that the inequality
 * $$\sqrt{p_{n+1}} - \sqrt{p_n} < 1 $$

holds for all $$n$$, where $$p_n$$ is the nth prime number. If $$g_n = p_{n+1} - p_n$$ denotes the nth prime gap, then Andrica's conjecture can also be rewritten as
 * $$g_n < 2\sqrt{p_n} + 1.$$

Empirical evidence
Imran Ghory has used data on the largest prime gaps to confirm the conjecture for $$n$$ up to 1.3002 × 1016. Using a table of maximal gaps and the above gap inequality, the confirmation value can be extended exhaustively to 4 × 1018.

The discrete function $$A_n = \sqrt{p_{n+1}}-\sqrt{p_n}$$ is plotted in the figures opposite. The high-water marks for $$A_n$$ 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.

Generalizations
As a generalization of Andrica's conjecture, the following equation has been considered:
 * $$ p _ {n+1} ^ x - p_ n ^ x = 1, $$

where $$ p_n $$ 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... which occurs for n = 30.

This conjecture has also been stated as an inequality, the generalized Andrica conjecture:
 * $$ p _ {n+1} ^ x - p_ n ^ x < 1 $$ for $$x < x_{\min}.$$