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

\( {\sqrt {p_{{n+1}}}}-{\sqrt {p_{n}}}<1 \)

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 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 \( x<x_{{\min }}. \)