We show that some partitions related to two of Ramanujan's mock theta functions are related to indefinite quadratic forms and real quadratic fields. In particular, we examine a third order mock theta function and a fifth order mock theta function.

Let p be an odd prime and let a be a positive integer. In this paper we investigate the sum $\genfrac{}{}{0pt}{}{{\sum }_{k=0}^{p^a-1}(h{p}^a-1}{\genfrac{}{}{0pt}{}{k\left)\right(2k}{k)/m^k\left(modp²\right)}}$, where h and m are p-adic integers with m ≢ 0 (mod p). For example, we show that if h ≢ 0 (mod p) and $p^a>3$, then $\genfrac{}{}{0pt}{}{{\sum }_{k=0}^{p^a-1}(h{p}^a-1}{\genfrac{}{}{0pt}{}{k\left)\right(2k}{{k)(-h/2)}^k\equiv ((1-2h)/\left(p^a\right))(1+h({(4-2/h)}^{p-1}-1))\left(modp²\right)}}$, where (·/·) denotes the Jacobi symbol. Here is another remarkable congruence: If $p^a>3$ then $\genfrac{}{}{0pt}{}{{\sum }_{k=0}^{p^a-1}(p^a-1}{\genfrac{}{}{0pt}{}{k\left)\right(2k}{{k)(-1)}^k\equiv 3^{p-1}(p^a/3)\left(modp²\right)}}$.

In this note, we estimate the distance between two $q$-nomial coefficients $\left|{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}_q-{\left(\genfrac{}{}{0pt}{}{n^{\text{'}}}{k^{\text{'}}}\right)}_q\right|$, where $(n,k)\ne (n^{\text{'}},k^{\text{'}})$ and $q\ge 2$ is an integer.