A Wolstenholme prime is a prime 
for which the central
binomial coefficient satisfies
 |
(1)
|
or, equivalently, for which
 |
(2)
|
where 
is the 
th
Bernoulli number and the congruence
is fractional.
Equivalently, a prime 
is a Wolstenholme prime when the harmonic
number satisfies
 |
(3)
|
A prime 
is a Wolstenholme prime iff
 |
(4)
|
where 
is the floor function and the congruence
is again fractional.
The only known Wolstenholme primes are 16843 and 2124679 (OEIS A088164). McIntosh (2004) reported no others up to 
.
Wolstenholme primes also supply composite counterexamples to two conjectured characterizations
of primes. Write 
in lowest terms and 
. For 
, 2, ..., the values of 
begin 0, 1, 5, 13, 77, 29, 223, 481, 4609, 4861, ... (OEIS
A064169). For 
, 4, ..., the values of 
begin 3, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13,
1, ... (OEIS A309391). If 
is a Wolstenholme prime, then
 |
(5)
|
Indeed, the terms of 
whose indices are divisible by 
sum to 
, which is 0 modulo 
. Among the remaining terms, pairing 
with 
shows that their sum through 
is 0 modulo 
; removing the final term gives
 |
(6)
|
It follows that 
is a composite number 
for which 
and 
. This disproves the conjecture
in OEIS A064169 that, for 
, 
holds iff 
is prime. It also disproves
the equivalent conjecture in OEIS A309391
that 
only when 
is prime (Adamczewski 2026). The counterexample
was found by an AI-generated proof formally verified in Lean (Epoch AI 2026).
-Adic Logarithm." Integers 20, Paper No. A42,
15 pp., 2020.