What's the name of this Powerful lemma? I have read a book,post this
Lemma:  Let $p$ be a prime then if $p-1 | k $ one has
$$\sum_{i=1}^{p} i^k \equiv -1 \pmod p$$
and if $p-1 \not | k$ then one has
$$\sum_{i=1}^{p} i^k \equiv 0 \pmod p$$
I can understand this lemma proof: 
proof :
The first assertion is obvious as each term will contribute $1$ except for $p^k$ which is $0$ and for the second, let the sum be $S$ and note that $g^kS \equiv S \pmod p$ where $g$ is a primitive root $\pmod p$. Since $g^k \not \equiv 1 \pmod p$ then one must have $S \equiv 0 \pmod p$.By done!
Question:I fell this lemma is very powefull,and  have other methods to prove this lemma?and what's the name of the lemma?or have some Application to paper?
 A: With high probability, this lemma doesn't have a name attributed to some author. One  may simply call it a "congruence for integer power sums".
See the 2010 article by Kieren MacMillan and Jonathan Sondow,   Proofs of Power Sum and Binomial Coefficient Congruences Via Pascal’s Identity , which is exclusively devoted to this lemma and still doesn't name if after somebody. 
In this article, there are references to many alternative methods of proof, e.g.   relying  on the theory of primitive roots, or invoking Lagrange’s theorem, or employing Bernoulli numbers and finite differences. 
Even more interestingly, this article presents a new elementary proof, using an identity for power sums proven by Pascal in 1654. This shows that technically it was possible that the lemma was known to Pascal or other people 350 years ago.
Further, the article also gives numerous references to applications, e.g. to prove theorems on Bernoulli numbers (Staudt-Clausen, Carlitz-von Staudt, Almkvist-Meurman) and to study the Erdös-Moser Diophantine equation as well as other exponential Diophantine equations and Stirling numbers of the second kind.
Also other standard volumes do not give the lemma a name, e.g. the classic textbook by G. H. Hardy and E. M. Wright,
An Introduction to the Theory of Numbers
, where it appears as theorem 119. This again supports that the lemma most likely does not have a name.
Also in here, the application is given to prove von Staudt's theorem.
