rational points on one dimensional sphere

Let

\mathbb{S}^{1}=\{(x,y)\in\mathbb{R}^{2}\ |\ x^{2}+y^{2}=1\}

be a one dimensional sphere.

We will denote by

\mathbb{S}^{1}_{\mathbb{Q}}=\{(x,y)\in\mathbb{Q}^{2}\ |\ (x,y)\in\mathbb{S}^{1}\}

the rational sphere. We shall try to describe $\mathbb{S}^{1}_{\mathbb{Q}}$ in terms of Pythagorean triplets.

Theorem. Let $(x,y)\in\mathbb{R}^{2}$ . Then $(x,y)\in\mathbb{S}^{1}_{\mathbb{Q}}$ if and only if there exists a Pythagorean triplet $a,b,c\in\mathbb{Z}$ (i.e. $|a|,|b|,|c|\in\mathbb{N}$ is a Pythagorean triplet) such that $x, y$ are of the form $\frac{a}{c}$ and $\frac{b}{c}$ .

Proof. ,, $\Leftarrow$ ” If (for example) $x=\frac{a}{c}$ and $y=\frac{b}{c}$ for a Pythagorean triplet $a,b,c\in\mathbb{Z}$ , then we have

x^{2}+y^{2}=\frac{a^{2}}{c^{2}}+\frac{b^{2}}{c^{2}}=\frac{a^{2}+b^{2}}{c^{2}}=% \frac{c^{2}}{c^{2}}=1

and thus $(x,y)\in\mathbb{S}^{1}_{\mathbb{Q}}$ .

,, $\Rightarrow$ ” Assume that $(x,y)\in\mathbb{S}^{1}_{\mathbb{Q}}$ . Then $x=\frac{p}{q}$ for some $p,q\in\mathbb{Z}$ . It follows, that

1=x^{2}+y^{2}=\frac{p^{2}}{q^{2}}+y^{2}

and this is if and only if

y=\frac{\sqrt{q^{2}-p^{2}}}{q}

(up to a sign of course). Therefore $y\in\mathbb{Q}$ if and only if $\sqrt{q^{2}-p^{2}}=n$ is an integer. In this case we have

x=\frac{p}{q},\ \ \ \ y=\frac{n}{q}.

Note, that

q^{2}-p^{2}=n^{2}

and thus

q^{2}=n^{2}+p^{2},

so $n,p,q\in\mathbb{Z}$ is a Pythagorean triplet, which completes the proof. $\square$

Corollary. The rational sphere $\mathbb{S}^{1}_{\mathbb{Q}}$ is an infinite set.

Proof. Let $m,n\in\mathbb{N}$ be natural numbers such that $n$ is fixed and even. Let $m$ run through primes. Then (due to the theorem in parent entry)

2mn,\ m^{2}-n^{2},\ m^{2}+n^{2}\in\mathbb{N}

is a Pythagorean triplet. Let

x_{m}=\frac{2mn}{m^{2}+n^{2}}=\frac{2n}{m+\frac{n^{2}}{m}}.

Ir follows from the theorem, that there exists $y_{m}\in\mathbb{Q}$ such that $(x_{m},y_{m})\in\mathbb{S}^{1}_{\mathbb{Q}}$ . It is easy to see, that $x_{m}=x_{m^{\prime}}$ if and only if $m=m^{\prime}$ and thus we generated infinitely many rational points on sphere. This completes the proof. $\square$

Title	rational points on one dimensional sphere
Canonical name	RationalPointsOnOneDimensionalSphere
Date of creation	2013-03-22 19:07:49
Last modified on	2013-03-22 19:07:49
Owner	joking (16130)
Last modified by	joking (16130)
Numerical id	7
Author	joking (16130)
Entry type	Definition
Classification	msc 11-00
Related topic	RationalSineAndCosine