The value of a given carry, when adding modulo p
(The Proof of (19))
Given integers n and m, we take r=n-m.
if there is a `carry' in the jth digit when we add m and r in base p; otherwise let
We observed, in the proof of Kummer's Theorem that, for each integer
Therefore, if we let
be the least residues, in absolute
value, of , respectively, so that times the
left side of (19), plus equals n-m-r = 0. However,
and (19) follows.