Andrica's conjecture
Andrica's conjecture (named after Dorin Andrica) is a conjecture regarding the gaps between prime numbers.[1]
The conjecture states that the inequality
- [math]\displaystyle{ \sqrt{p_{n+1}} - \sqrt{p_n} \lt 1 }[/math]
holds for all [math]\displaystyle{ n }[/math], where [math]\displaystyle{ p_n }[/math] is the nth prime number. If [math]\displaystyle{ g_n = p_{n+1} - p_n }[/math] denotes the nth prime gap, then Andrica's conjecture can also be rewritten as
- [math]\displaystyle{ g_n \lt 2\sqrt{p_n} + 1. }[/math]
Empirical evidence
Imran Ghory has used data on the largest prime gaps to confirm the conjecture for [math]\displaystyle{ n }[/math] 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 [math]\displaystyle{ A_n = \sqrt{p_{n+1}}-\sqrt{p_n} }[/math] is plotted in the figures opposite. The high-water marks for [math]\displaystyle{ A_n }[/math] 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.
Generalizations
As a generalization of Andrica's conjecture, the following equation has been considered:
- [math]\displaystyle{ p _ {n+1} ^ x - p_ n ^ x = 1, }[/math]
where [math]\displaystyle{ p_n }[/math] is the nth prime and x can be any positive number.
The largest possible solution x is easily seen to occur for [math]\displaystyle{ n=1 }[/math], when xmax = 1. The smallest solution 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:
- [math]\displaystyle{ p _ {n+1} ^ x - p_ n ^ x \lt 1 }[/math] for [math]\displaystyle{ x \lt x_{\min}. }[/math]
See also
References and notes
- Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. ISBN 978-0-387-20860-2.
External links
- Andrica's Conjecture at PlanetMath
- Generalized Andrica conjecture at PlanetMath
- Weisstein, Eric W.. "Andrica's Conjecture". http://mathworld.wolfram.com/AndricasConjecture.html.
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