Wolstenholme's Theorem

Theorem

Let $p$ be a prime number such that $p \ge 5$.

Then:

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

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

Proof

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Also see

• Wolstenholme's Conjecture

• Babbage's Congruence

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)