Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

In Wikipedia entry on Wolstenholme's theorem, it says

The second formulation $\binom{ap}{bp}=\binom{a}{b}\pmod{p^3}$ of Wolstenholme's theorem is due to J. W. L. Glaisher.

But there is no reference to it. Does anyone know which paper of J. W. L. Glaisher contains the above statement?

share|improve this question
Did you try Dickson's "Theory of Numbers"? –  Péter Komjáth Apr 23 '13 at 5:08
...or Google: the first Google hit for glaisher wolstenholme is <arxiv.org/pdf/1111.3057.pdf>;, where on page 25 there are three references ([32] to [34]) to papers by Glaisher published in the Q.J.Math. in 1900 or 1901; the second of these ("On the residues of the sums of products of the first $p - 1$ numbers, and their powers, to modulus $p^2$ or $p^3$") seems particularly promising. –  Noam D. Elkies Apr 23 '13 at 5:27
Thanks. I have added these references to the Wikipedia page. –  Z.H. Apr 23 '13 at 9:09

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.