The value of a given carry, when adding modulo p

(The Proof of (19))


Proof: Given integers n and m, we take r=n-m. Define if there is a `carry' in the jth digit when we add m and r in base p; otherwise let (including ). 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.