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 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
p
n
+
1.
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
=
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 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.
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
.
