Let $\sigma_p(m,n)$ (resp. $\pi_p(m,n)$) denote the sum of the carries when adding (resp. multiplying) the numbers $m=\sum_{k\ge0}m_kp^k$ and $n=\sum_{k\ge0}n_kp^k$ using base-$p$ arithmetic where $m_k,n_k\in\{0,\dots,p-1\}$ for $k\ge0$. Let $s_p(m)=\sum_{k\ge0}m_k$ be the sum of the base-$p$ digits.

**Question 1.** Does someone know a reference for the following (cohomological) formulas?
$$\sigma_p(m,n)=\frac{s_p(m)+s_p(n)-s_p(m+n)}{p-1},\quad \pi_p(m,n)=\frac{s_p(m)s_p(n)-s_p(mn)}{p-1}.$$

**Question 2.** Does someone know what $\pi_p(m,n)$ counts (besides the number of carries)?

I will happily type my (short) proofs if there is interest, and there is no obvious reference. A famous result of Kummer says that $\sigma_p(m,n)$ is the exponent of the largest power of $p$ dividing $\binom{m+n}{n}$. (The formula for $\pi_p(m,n)$ arose from a problem in finite geometry.)

Edit: Typo corrected ($\alpha$ changed to $\sigma$). Thanks R. Also, replaced `number of carries' with 'sum of carries' as $c_k>1$ happens, see proof below.