### - Art Gallery -

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

The conjecture states that the inequality

$${\sqrt {p_{{n+1}}}}-{\sqrt {p_{n}}}<1$$

holds for all n, where $$p_{n}$$ is the nth prime number. If $$g_{n}=p_{{n+1}}-p_{n}$$ denotes the nth prime gap, then Andrica's conjecture can also be rewritten as

$$g_{n}<2{\sqrt {p_{n}}}+1.$$

Empirical evidence

Imran Ghory has used data on the largest prime gaps to confirm the conjecture for n up to 1.3002 × 1016. Using a table of maximal gaps and the above gap inequality, the confirmation value can be extended exhaustively to 4 × 1018.

The discrete function $$A_{n}={\sqrt {p_{{n+1}}}}-{\sqrt {p_{n}}}$$ is plotted in the figures opposite. The high-water marks for $$A_{n}$$ 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:

$$p_{{n+1}}^{x}-p_{n}^{x}=1,$$

where $$p_{n}$$ is the nth prime and x can be any positive number.

The largest possible solution x is easily seen to occur for n=1, 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:

$$p_{{n+1}}^{x}-p_{n}^{x}<1$$for $$x<x_{{\min }}.$$

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

References and notes

Andrica, D. (1986). "Note on a conjecture in prime number theory". Studia Univ. Babes–Bolyai Math. 31 (4): 44–48. ISSN 0252-1938. Zbl 0623.10030.

Prime Numbers: The Most Mysterious Figures in Math, John Wiley & Sons, Inc., 2005, p. 13.

Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. ISBN 978-0-387-20860-2. Zbl 1058.11001.

Andrica's Conjecture at PlanetMath
Generalized Andrica conjecture at PlanetMath
Weisstein, Eric W. "Andrica's Conjecture". Mathworld.