Editing Floor and ceiling functions (section)

===Quadratic reciprocity===
Gauss's third proof of [[quadratic reciprocity]], as modified by Eisenstein, has two basic steps.<ref>Lemmermeyer, § 1.4, Ex. 1.32–1.33</ref><ref>Hardy & Wright, §§ 6.11–6.13</ref>

Let ''p'' and ''q'' be distinct positive odd prime numbers, and let <math>m = \tfrac12(p - 1),</math>  <math>n = \tfrac12(q - 1).</math>

First, [[Gauss's lemma (number theory)|Gauss's lemma]] is used to show that the [[Legendre symbol]]s are given by

:<math>\begin{align}
\left(\frac{q}{p}\right)
&= (-1)^{\left\lfloor\frac{q}{p}\right\rfloor + \left\lfloor\frac{2q}{p}\right\rfloor +
   \dots + \left\lfloor\frac{mq}{p}\right\rfloor }, \\[5mu]
\left(\frac{p}{q}\right)
&= (-1)^{\left\lfloor\frac{p}{q}\right\rfloor + \left\lfloor\frac{2p}{q}\right\rfloor + 
   \dots + \left\lfloor\frac{np}{q}\right\rfloor }.
\end{align}</math>

The second step is to use a [[Geometric series|geometric]] argument to show that

:<math>\left\lfloor\frac{q}{p}\right\rfloor +\left\lfloor\frac{2q}{p}\right\rfloor +\dots +\left\lfloor\frac{mq}{p}\right\rfloor

+\left\lfloor\frac{p}{q}\right\rfloor +\left\lfloor\frac{2p}{q}\right\rfloor +\dots +\left\lfloor\frac{np}{q}\right\rfloor

= mn.
</math>

Combining these formulas gives quadratic reciprocity in the form

:<math>\left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{mn}=(-1)^{\frac{p-1}{2}\frac{q-1}{2}}.</math>

There are formulas that use floor to express the quadratic character of small numbers mod odd primes ''p'':<ref>Lemmermeyer, p. 25</ref>
:<math>\begin{align}
\left(\frac{2}{p}\right) &= (-1)^{\left\lfloor\frac{p+1}{4}\right\rfloor}, \\[5mu]
\left(\frac{3}{p}\right) &= (-1)^{\left\lfloor\frac{p+1}{6}\right\rfloor}.
\end{align}</math>