# Sum of binomials

• March 3rd 2010, 11:58 AM
james_bond
Sum of binomials
Let $p>3$ be a prime. $\sum_{i=1}^p\binom {i\cdot p}{p}\cdot\binom {\left(p-i+1\right)p}{p}\equiv ?\mod p^2$.
• March 3rd 2010, 07:30 PM
NonCommAlg
Quote:

Originally Posted by james_bond
Let $p>3$ be a prime. $\sum_{i=1}^p\binom {i\cdot p}{p}\cdot\binom {\left(p-i+1\right)p}{p}\equiv ?\mod p^2$.

the answer is $\frac{p(p+1)(p+2)}{6}$ and we don't need the condition $p > 3.$ to prove this let $f(x)=\prod_{k=1}^{p-1}(x-k)=x^{p-1}-a_1x^{p-2} + \cdots -a_{p-2}x + a_{p-1}.$

by the Fermat's little theorem, in $(\mathbb{Z}/p\mathbb{Z})[x],$ we have $f(x)=x^{p-1}-1.$ as a result $(p-1)!=a_{p-1} \equiv -1 \mod p,$ which is Wilson's theorem, and

$a_j \equiv 0 \mod p,$ for all $j \neq p-1.$ thus, for any $n \in \mathbb{N}: \ \binom{np}{p}=\frac{np(np-1) \cdots (np-p+1)}{p!}=\frac{nf(np)}{(p-1)!} \equiv n \mod p^2.$ so your sum, modulo $p^2$,

is equal to $\sum_{i=1}^{p}i(p-i+1)=\frac{p(p+1)(p+2)}{6}. \ \Box$

Remark 1. this method gives another proof for the problem you asked in here.

Remark 2. there's a theorem (i don't remember its name) which says that for any prime number $p > 3: \ a_{p-2} \equiv 0 \mod p^2.$ does anybody know the

name of this theorem? anyway, using this result we get this stronger result that, modulo $p^3,$ your sum for $p > 3$ is still equal to $\frac{p(p+1)(p+2)}{6}.$
• March 3rd 2010, 08:54 PM
NonCommAlg
Quote:

Originally Posted by NonCommAlg

Remark 2. there's a theorem (i don't remember its name) which says that for any prime number $p > 3: \ a_{p-2} \equiv 0 \mod p^2.$ does anybody know the name of this theorem?

ok, i found it! it's Wolstenholme's theorem.
• March 3rd 2010, 10:11 PM
Drexel28
Quote:

Originally Posted by NonCommAlg
ok, i found it! it's Wolstenholme's theorem.

I can't imagine that's a particularly commonly used theorem.
• March 4th 2010, 07:30 PM
NonCommAlg
Quote:

Originally Posted by Drexel28
I can't imagine that's a particularly commonly used theorem.

well, it's a non-trivial result and that's good enough to make it useful!