Andrica's conjecture

(a) The function

A_{n}

for the first 100 primes.

(b) The function

A_{n}

for the first 200 primes.

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. It is conjectured that 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.^[1]

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.^[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 $A_{n}={\sqrt {p_{n+1}}}-{\sqrt {p_{n}}}$ 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 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... (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 }.

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.
^ Wells, David (May 18, 2005). Prime Numbers: The Most Mysterious Figures in Math. Hoboken (N.J.): Wiley. p. 13. ISBN 978-0-471-46234-7.