# Math Help - Sum of powers

1. ## Sum of powers

Hi verybody,
Is there a proof of:

Sum[(i^s - 1), {i, 1, a}] mod( a ) = Sum[(i - 1)^s, {i, 1, a}] mod(
a )
a, s positive integers (could be equal).
If there is, please tell me.

2. Are you saying

$\sum_{i=1}^a(i^s-1) \equiv \sum_{i=1}^a(i-1)^s \mod a$

?

3. exactly (a>s)

4. $
\sum_{i=1}^a(i^s-1) \equiv \Big(\sum_{i=1}^ai^s\Big)-a \equiv \sum_{i=1}^ai^s \mod a$

and
$\sum_{i=1}^ai^s \equiv \sum_{i=1}^a(i-1)^s \mod a$

because both sums are taken over complete residue systems $\mod a$. Works also if $s>a$ b.t.w.