Proof
(related to Lemma: Reciprocity Law for Floor Functions)
- By hypothesis, $p,q\in\mathbb Z$ are co-prime, odd, and both $ > 2.$
- We consider the numbers $lp-kq$ for $k=1,\ldots,\frac{p-1}{2}$ and $l=1,\ldots,\frac{q-1}{2}.$
- None of these numbers equals $0$, otherwise we would have that $lp=kq$ and $q$ would be a divisor of $lp$ and since $q$ and $p$ are distinct, $q$ would divide $l$, which is impossible due to $l < \frac q2.$
- According to the creation of reduced residue systems from others, these numbers must therefore be all distinct.
- Therefore, there are $\frac{p-1}{2}\cdot\frac{q-1}{2}$ different numbers $lp-kq$ for $k=1,\ldots,\frac{p-1}{2}$ and $l=1,\ldots,\frac{q-1}{2}.$
- Since for $l=1,\ldots,\frac{q-1}{2}$ we have $$k < \frac{lp}q,\quad k=1,\ldots,\frac{p-1}{2},$$ and since $\frac{lp}q$ is not an integer, there are exactly $\left\lfloor\frac{lp}q\right\rfloor$ solutions of $ k=1,\ldots,\frac{p-1}{2},$ using number of multiples of a given number less than another number.
- Therefore, there are $$\sum_{l=1}^{\frac{q-1}{2}}\left\lfloor\frac{lp}q\right\rfloor$$ positive integers $lp-kq$ for $k=1,\ldots,\frac{p-1}{2}$ and $l=1,\ldots,\frac{q-1}{2}.$
- Since $k < \frac{\frac{q}{2}p}{q}=\frac p2,$ for symmetry reasons, we have $$\sum_{k=1}^{\frac{p-1}{2}}\left\lfloor\frac{kq}p\right\rfloor$$ negative integers $lp-kq$ for $k=1,\ldots,\frac{p-1}{2}$ and $l=1,\ldots,\frac{q-1}{2}.$
- Altogether, we have $$\frac{p-1}{2}\cdot\frac{q-1}{2}=\sum_{l=1}^{\frac{q-1}{2}}\left\lfloor\frac{lp}q\right\rfloor+\sum_{k=1}^{\frac{p-1}{2}}\left\lfloor\frac{kq}p\right\rfloor.$$
∎
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References
Bibliography
- Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927
