Base-p digit sum ($d_p(n)$)

For a nonnegative integer $n$ and integer base $p\ge 2$, write its base-p expansion asThe base-$p$ digit sum is $d_p(n)={\sum }_r{\alpha }_r$, with $d_p(0)=0$. When $p$ is a prime number, the Legendre formula becomesTo see the connection, divide the base-p expansion by $p^j$ and discard the fractional part. Swapping the resulting finite sums and using a finite geometric series givesIn particular, $50=1\cdot 27+2\cdot 9+1\cdot 3+2=(1212{)}_3$, so $d_3(50)=6$ and $v_3(50!)=(50-6)/2=22$. The leading numeral digit contributes thirteen, the next contributes eight, the next one, and the units contribute zero. Thus the base-p digit sum compresses the same contribution ledger into information already present in positional notation.

For nonnegative integers $x,y$ and an integer base $p\ge 2$,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.