### Archive

Posts Tagged ‘Legendre’

Prove that if $p \geq 3$ is a prime number, then any natural divisor of $\left\lfloor \frac{p+1}{4} \right\rfloor$ is a quadratic residue modulo $p$.

## Sum of integer parts and prime number

Let $p$ be a prime such that $p \equiv 1(\text{mod } 4)$. Calculate the sum
$\displaystyle s=\left\lfloor \sqrt{1\cdot p}\right\rfloor + \left\lfloor \sqrt{2\cdot p}\right\rfloor+\cdots +\left\lfloor \sqrt{\frac{p-1}{4}\cdot p}\right\rfloor$.
IMO

## Quadratic residues and Legendre’s Symbol

April 2, 2010 1 comment

In this wikipedia article you can find a bit of history, and the properties of Legendre’s Symbol. It is used many times in number theory problems, and is very helpful.

It is defined as $\displaystyle \left(\frac{a}{p}\right)= \begin{cases} 1 & \text{ if } a\equiv x^2 ( \text{mod } p) \\ 0 & \text{otherwise} \end{cases}$, where $a \in \mathbb{Z}$ and $p$ is a prime not dividing $a$.

Given a prime number $p\equiv 7 (mod\ 8)$, evaluate $\displaystyle \sum_{k=1}^{(p-1)/2} \left\lfloor \frac{k^2}{p}+\frac{1}{2} \right\rfloor$.