HOME

TheInfoList



OR:

Andrica's conjecture (named after Romanian mathematician Dorin Andrica ( es)) is a
conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
regarding the gaps between
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s. 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 A prime gap is the difference between two successive prime numbers. The ''n''-th prime gap, denoted ''g'n'' or ''g''(''p'n'') is the difference between the (''n'' + 1)-st and the ''n''-th prime numbers, i.e., :g_n = p_ - p_n. ...
, 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 PlanetMath is a free content, free, collaborative, mathematics online encyclopedia. Intended to be comprehensive, the project is currently hosted by the University of Waterloo. The site is owned by a US-based nonprofit corporation, "PlanetMath.org ...

''Generalized Andrica conjecture''
at
PlanetMath PlanetMath is a free content, free, collaborative, mathematics online encyclopedia. Intended to be comprehensive, the project is currently hosted by the University of Waterloo. The site is owned by a US-based nonprofit corporation, "PlanetMath.org ...
* https://drive.google.com/file/d/1jWbjCbTn5Tf0twXJJochRt7ONwqs6l3D/view?usp=sharing {{Prime number conjectures Conjectures about prime numbers Unsolved problems in number theory