Andrica's conjecture (named after Romanian mathematician Dorin Andrica) is a conjecture regarding the gaps between prime numbers.[1]
The conjecture states that the inequality
holds for all n {\displaystyle n} , where p n {\displaystyle p_{n}} is the nth prime number. If g n = p n + 1 − − --> p n {\displaystyle g_{n}=p_{n+1}-p_{n}} 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 n {\displaystyle n} up to 1.3002 × 1016.[2] 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 = p n + 1 − − --> p n {\displaystyle A_{n}={\sqrt {p_{n+1}}}-{\sqrt {p_{n}}}} is plotted in the figures opposite. The high-water marks for A n {\displaystyle 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.
As a generalization of Andrica's conjecture, the following equation has been considered:
where p n {\displaystyle 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... (sequence A038458 in the OEIS) which occurs for n = 30.
This conjecture has also been stated as an inequality, the generalized Andrica conjecture:
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