rational points on one dimensional sphere



be a one dimensional sphere.

We will denote by


the rational sphere. We shall try to describe 𝕊1 in terms of Pythagorean triplets.

Theorem. Let (x,y)2. Then (x,y)𝕊1 if and only if there exists a Pythagorean triplet a,b,c (i.e. |a|,|b|,|c| is a Pythagorean triplet) such that x,y are of the form ac and bc.

Proof. ,,” If (for example) x=ac and y=bc for a Pythagorean triplet a,b,c, then we have


and thus (x,y)𝕊1.

,,” Assume that (x,y)𝕊1. Then x=pq for some p,q. It follows, that


and this is if and only if


(up to a sign of course). Therefore y if and only if q2-p2=n is an integer. In this case we have


Note, that


and thus


so n,p,q is a Pythagorean triplet, which completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof.

Corollary. The rational sphere 𝕊1 is an infinite setMathworldPlanetmath.

Proof. Let m,n be natural numbersMathworldPlanetmath such that n is fixed and even. Let m run through primes. Then (due to the theorem in parent entry)


is a Pythagorean triplet. Let


Ir follows from the theorem, that there exists ym such that (xm,ym)𝕊1. It is easy to see, that xm=xm if and only if m=m and thus we generated infinitely many rational points on sphere. This completes the proof.

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