Wolstenholme's Theorem

If $p$ is a Prime $>3$, then the Numerator of

$$1+{\textstyle{1\over 2}}+{\textstyle{1\over 3}}+\ldots+{\textstyle{1\over p-1}}$$

is divisible by $p^2$ and the Numerator of

$$1+{1\over 2^2}+{1\over 3^2} +\ldots+{1\over(p-1)^2}$$

is divisible by $p$. These imply that if $p\geq 5$ is Prime, then

$${2p-1\choose p-1} \equiv 1\ ({\rm mod\ } p^3).$$

References

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 85, 1994.

Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, p. 21, 1989.