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

The conjecture states that the inequality

p
n
+
1
−
p
n
<
1
holds for all
n
, where
p
n
is the *n*th prime number. If
g
n
=
p
n
+
1
−
p
n
denotes the *n*th prime gap, then Andrica's conjecture can also be rewritten as

g
n
<
2
p
n
+
1.
Imran Ghory has used data on the largest prime gaps to confirm the conjecture for
n
up to 1.3002 × 10^{16}. Using a table of maximal gaps and the above gap inequality, the confirmation value can be extended exhaustively to 4 × 10^{18}.

The discrete function
A
n
=
p
n
+
1
−
p
n
is plotted in the figures opposite. The high-water marks for
A
n
occur for *n* = 1, 2, and 4, with *A*_{4} ≈ 0.670873..., with no larger value among the first 10^{5} 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.

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 *n*th 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 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
.