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
:
holds for all
, where
is the ''n''th prime number. If
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
:
Empirical evidence
Imran Ghory has used data on the largest prime gaps to confirm the conjecture for
up to .
Using a more recent table of
maximal gaps, the confirmation value can be extended exhaustively to > 2
64.
The discrete function
is plotted in the figures opposite. The high-water marks for
occur for ''n'' = 1, 2, and 4, with ''A''
4 ≈ 0.670873..., with no larger value among the first 10
5 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
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:
:
for
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