Andrica's conjecture explained
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
} - \sqrt < 1 holds for all
, where
is the
nth prime number. If
denotes the
nth
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
up to 1.3002 × 10
16.
[2] Using a table of maximal gaps and the above gap inequality, the confirmation value can be extended exhaustively to 4 × 10
18.
The discrete function
}-\sqrt is plotted in the figures opposite. The high-water marks for
occur for
n = 1, 2, and 4, with
A4 ≈ 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
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... which occurs for n = 30.
This conjecture has also been stated as an inequality, the generalized Andrica conjecture:
for
See also
References and notes
- Book: Guy, Richard K. . Richard K. Guy . Unsolved problems in number theory . . 3rd . 2004 . Section A8. 978-0-387-20860-2 . 1058.11001 .
External links
Notes and References
- D. . Andrica . Note on a conjecture in prime number theory . Studia Univ. Babes–Bolyai Math. . 31 . 1986 . 4 . 44–48 . 0623.10030 . 0252-1938 .
- Book: Wells, David . Prime Numbers: The Most Mysterious Figures in Math . Wiley . Hoboken (N.J.) . May 18, 2005 . 978-0-471-46234-7 . 13.
