On binomial coefficients modulo squares of primes

Abstract

We give elementary proofs for the Apagodu-Zeilberger-Stanton-Amdeberhan-Tauraso congruences $$\sum _{n=0}^{p-1}\left(\genfrac{}{}{0ex}{}{2n}{n}\right)\equiv {\eta }_p\mathrm{mod}{p}^2,$$ $$\sum _{n=0}^{rp-1}\left(\genfrac{}{}{0ex}{}{2n}{n}\right)\equiv {\eta }_p\sum _{n=0}^{r-1}\left(\genfrac{}{}{0ex}{}{2n}{n}\right)\mathrm{mod}{p}^2$$ and $$\sum _{n=0}^{rp-1}\sum _{m=0}^{sp-1}{\left(\genfrac{}{}{0ex}{}{n+m}{m}\right)}^2\equiv {\eta }_p\sum _{m=0}^{r-1}\sum _{n=0}^{s-1}{\left(\genfrac{}{}{0ex}{}{n+m}{m}\right)}^2\mathrm{mod}{p}^2,$$ where $p$ is an odd prime, $r$ and $s$ are nonnegative integers, and ${\eta }_p=\{\begin{array}{ll}0,& \text{if }p\equiv 0\mathrm{mod}3;\\ 1,& \text{if }p\equiv 1\mathrm{mod}3;\\ -1,& \text{if }p\equiv 2\mathrm{mod}3\end{array}.$$

27 pages. Semi-expository. Ancillary file is an archived version of the [Grinbe17b] reference, entirely expository. Comments are welcome! v2 corrects a few typos and updates references

References in corpus (3)

• Notes on the combinatorial fundamentals of algebra
• Lucas' theorem: its generalizations, extensions and applications (1878--2014)
• On two conjectural supercongruences of Apagodu and Zeilberger

Cited by in corpus (1)

• Uniformizer of the False Tate Curve Extension of ${\mathbb{Q}}_p$