Babbage's Congruence

Theorem

Let $p$ be a prime number such that $p \ge 3$.

Let $a, b \in \Z_{\ne 0}$ be non-zero integers.

Then:

$\dbinom {a p} {b p} \equiv \dbinom a b \pmod {p^2}$

where $\dbinom a b$ denotes a binomial coefficient.

Proof

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Also see

• Wolstenholme's Theorem

Source of Name

This entry was named for Charles Babbage.

Sources

• 1819: Charles Babbage: Demonstration of a theorem relating to prime numbers (The Edinburgh Philosophical Journal Vol. 1: pp. 46 – 49)