Andrica's conjecture

(a) The function $A_n$ for the first 100 primes.

(b) The function $A_n$ for the first 200 primes.

(c) The function $A_n$ for the first 500 primes.

Graphical proof for Andrica's conjecture for the first (a)100, (b)200 and (c)500 prime numbers. The function $A_n$ is always less than 1.

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 x 10¹⁶. Using a table of maximal gaps and the above gap inequality, the confirmation value can be extended exhaustively to 4 x 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 x_max=1. The smallest solution x is conjectured to be x_min ≈ 0.567148... (sequence A038458 in 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 $x < x_{\min}.$

See also

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

References and notes

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

External links

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