Let $p > 2$ be an odd and fixed prime number. The Legendre symbol $p$ $\left(\frac np\right)$ is an arithmetic function defined using the quadratic residues modulo $p$ as follows: $$\left(\frac np\right):=\begin{cases} 1&\text{if }n\text{ is quadratic residue modulo }p\text{ and }p\not\mid n,\\ -1&\text{if }n\text{ is a quadratic nonresidue modulo }p\text{ and }p\not\mid n,\\ 0&\text{if }p\mid n. \end{cases}$$
Definitions: 1
Explanations: 2
Lemmas: 3
Proofs: 4 5 6 7 8 9 10
Propositions: 11 12 13 14
Sections: 15 16
Solutions: 17 18
Theorems: 19 20 21 22
