Andrica's conjecture
Andrica's conjecture (named after Dorin Andrica) is a conjecture regarding the gaps between prime numbers. The conjecture states that the inequality holds for all formula_2, where formula_3 is the "n"th prime number. If formula_4 denotes the "n"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 formula_2 up to 1.3002 × 10. Using a table of maximal gaps and the above gap inequality, the confirmation value can be extended exhaustively to 4 × 10. The discrete function  formula_7 is plotted in the figures opposite. The high-water marks for formula_8 occur for "n" = 1, 2, and 4, with "A" ≈ 0.670873..., with no larger value among the first 10 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: where formula_10 is the "n"th prime and "x" can  be any positive number. The largest possible solution for "x" is easily seen to occur for "n"=1, when "x" = 1. The smallest solution for "x" is conjectured to be "x" ≈ 0.567148... which occurs for "n" = 30. This conjecture has also been stated as an inequality, the generalized Andrica conjecture: