Andrica's conjecture

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

The conjecture states that the inequality
:$\sqrt - \sqrt < 1$
holds for all $n$, where $p_n$ is the ''n''th prime number. If $g_n = p_ - p_n$ denotes the ''n''th prime gap, then Andrica's conjecture can also be rewritten as
:$g_n < 2\sqrt + 1.$

Empirical evidence

Imran Ghory has used data on the largest prime gaps to confirm the conjecture for $n$ up to . Using a more recent table of maximal gaps, the confirmation value can be extended exhaustively to > 2⁶⁴.

The discrete function $A_n = \sqrt-\sqrt$ is plotted in the figures opposite. The high-water marks for $A_n$ occur for ''n'' = 1, 2, and 4, with ''A''₄ ≈ 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:
:$p _ ^ x - p_ n ^ x = 1,$
where $p_n$ 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''_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:
:$p _ ^ x - p_ n ^ x < 1$ for $x < x_.$

See also

* Cramér's conjecture
* Legendre's conjecture
* Firoozbakht's conjecture

References and notes

*

External links

''Andrica's Conjecture''
at PlanetMath

''Generalized Andrica conjecture''
at PlanetMath
* https://drive.google.com/file/d/1jWbjCbTn5Tf0twXJJochRt7ONwqs6l3D/view?usp=sharing

{{Prime number conjectures

Conjectures about prime numbers
Unsolved problems in number theory