holds, where the discrete function is plotted above. The high-water marks for occur for , 2, and 4, with , with no larger value among
the first primes. Since the Andrica function falls asymptotically
as
increases, a prime gap of ever increasing size is needed
to make the difference large as becomes large. It therefore seems highly likely the conjecture
is true, although this has not yet been proven.

bears a strong resemblance to the prime difference
function, plotted above, the first few values of which are 1, 2, 2, 4, 2, 4,
2, 4, 6, 2, 6, ... (OEIS A001223).

A generalization of Andrica's conjecture considers the equation

and solves for .
The smallest such
is
(OEIS A038458), known as the Smarandache
constant, which occurs for and (Perez).