Base-p digit sum ( $d_{p} (n)$ )

If

n = \sum_{r \geq 0} α_{r} p^{r}, 0 \leq α_{r} < p,

then the base-

p

d_{p} (n) = \sum_{r} α_{r}

. The Legendre formula can equivalently be written

v_{p} (n!) = \frac{n - d _{p} ( n )}{p - 1} .

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- Subadditivity of the base-p digit sum Base-p digit sum

Subadditivity of the base-p digit sum

For nonnegative integers

x

and

y

,

d_{p} (x + y) \leq d_{p} (x) + d_{p} (y) .

In column addition, every carry replaces

p

units in one digit by one unit in the next digit and therefore decreases the digit sum by

p - 1

.

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