Evaluation rules (& names)
Suppose \(p\) is an odd prime. Then:
EvalZero: $$\left(\frac{0}{p}\right) = 0$$
EvalOne: \(1^2 \equiv 1 \pmod{p}\), so $$\left(\frac{1}{p}\right) = 1$$
Modulo: If \(a \equiv b \pmod{p}\), then $$\left(\frac{a}{p}\right) = \left(\frac{b}{p}\right)$$
Multiplicative: For all \(a, b\), we have $$\left(\frac{ab}{p}\right) = \left(\frac{a}{p}\right)\left(\frac{b}{p}\right)$$
See Lemma 15.13 in the notes.
Power: (This is really just a consequence of
Multiplicative). For all \(a\), we have
$$\left(\frac{a^k}{p}\right) = \left(\frac{a}{p}\right)^k$$
In particular,
$$\left(\frac{a^k}{p}\right) = \begin{cases}
1 & \mathrm{if} \; k \; \mathrm{is} \; \mathrm{even} \\
\left(\frac{a}{p}\right) & \mathrm{if} \; k \; \mathrm{is} \; \mathrm{odd}
\end{cases}$$
See Lemma 15.13 in the notes.
EvalMinusOne: $$\left(\frac{p-1}{p}\right) =
\begin{cases}
1 & \mathrm{if} \; p \equiv 1 \pmod{4} \\
-1 & \mathrm{if} \; p \equiv -1 \pmod{4} \\
\end{cases}$$
See Theorem 15.11 in the notes.
EvalTwo: $$\left(\frac{2}{p}\right) =
\begin{cases}
1 & \mathrm{if} \; p \equiv \pm 1 \pmod{8} \\
-1 & \mathrm{if} \; p \equiv \pm 3 \pmod{8} \\
\end{cases}$$
See Proposition 16.9 in the notes.
Reciprocity: Let \(q\) be an odd prime distinct from \(p\).
If \(p \equiv q \equiv -1 \pmod{4}\), then
$$\left(\frac{p}{q}\right) = -\left(\frac{q}{p}\right)$$
Otherwise,
$$\left(\frac{p}{q}\right) = \left(\frac{q}{p}\right)$$
See Theorem 16.1 in the notes.
What is this?
This was a fun little app I put together while revising for a number
theory exam. The core idea is that the app helps you teach yourself to
determine whether a given number \(y\) is a quadratic residue
modulo some odd prime \(p\), that is, if there exists \(x\) such that
\(x^2 \equiv y \pmod{p}\).
The Legendre symbol \(\left(\frac{y}{p}\right)\) is defined to be \(0\)
if \(y\) is a multiple of \(p\), \(1\) if \(y\) is a quadratic residue
modulo \(p\), and \(-1\) otherwise.
The notes referred to within the evaluation rules are the set of
lecture notes which were used in the 2017/18 Introduction to Number
Theory course at the University of Edinburgh. If you don't have those
notes, a little googling will hopefully help you find proofs of these
facts. Alternatively, you might try to prove them yourself!