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Posts Tagged ‘Legendre’

Quadratic residue divisor

April 5, 2010 Leave a comment

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

April 5, 2010 Leave a comment

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.

For further properties, please refer to the article linked.

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Sum of integer parts

March 31, 2010 Leave a comment

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.
AMM 10852 Read more…

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