Wolstenholme's Conjecture

Conjecture

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Suppose that:

$\dbinom {2 n - 1} {n - 1} \equiv 1 \pmod {n^3}$

where $\dbinom {2 n - 1} {n - 1}$ denotes a binomial coefficient.

Then $n$ is a prime number

Also see

• Wolstenholme's Theorem, of which this is the converse

Source of Name

This entry was named for Joseph Wolstenholme.

Sources

• 1862: Joseph Wolstenholme: On certain properties of prime numbers (Quart. J. Pure Appl. Math. Vol. 5: pp. 35 – 39)