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 the congruence is again fractional.
The only known Wolstenholme primes are 16843 and 2124679 (OEIS A088164). There are no others up to 
(McIntosh 2004).
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.