= Subadditivity of the base-p digit sum
For <nonnegative integers> $x,y$ and an <integer> base $p\geq2$,
$$
d_p(x+y)\leq d_p(x)+d_p(y).
$$
In column <addition>, every <arithmetic carry> replaces $p$ units in one <numeral digit> by one unit in the next position and therefore decreases the <base-p digit sum> by $p-1$. Count each outgoing carry separately, including each step of a chain. There is equality precisely when there are no carries. For <prime number> bases, <Kummer's theorem> turns this digit-sum deficit into a <prime exponent>.
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