Number Theory | Rational Points on the Unit Circle

We describe all points on the unit circle with rational coordinates. Furthermore, we outline a strategy for finding rational points on other quadratic curves.

[georgesadler]

Professor Penn, thank you for a monster analysis on the Rational Points on a Circle.

[lexinwonderland]

this video feels like the Mecca of Dr. Penn's videos, it's where it all began, all the new viewers should come pay pilgrimage. also the editing involved in backflips AND blackboard changes, holy wow!!!

[mdperpe]

An easy way to see that there are rational points on the unit circle is to take Pythagorean triples (a, b, c), i.e. a² + b² = c². Then the points (±a/c, ±b/c) are on the unit circle.
Furthermore, if (x1, y1) and (x2, y2) are rational points on the unit circle, then so is (x1 x2 - y1 y2, x1 y2 + x2 y1).

[lukewaite]

bro nice watched you for a while but never seen your backflip era before

[damianbla]

Finally I have found a video from pre-"and that's a good place to stop" era ;)

[lexinwonderland]

Also, (I know this video is ancient, but) professor, could you do more videos like this but on elliptic curves? I know they tie in with group theory among other things that i want to study, but I would really like to see your explanation of the fundamentals so the connection with modular forms makes more sense. Thanks!

[waynemv]

At 8:03, should we assume that a, b, c, d, and e are integers, or that they are rational?

[SpeedcoreDancecore]

Can a quadratic curve also include mixed terms? ax² + bx + cxy + dy + ey² + f = 0

[زكريا_حسناوي]

لقد أثبتتُ النظرية من اجل دائرة الوحدة باستعمال ثلاثيات فيثاغورس كما يلي:
ليكن a^2+b^2+=c^2، نعطي x=a/c وبالتالي y=sqrt(1--(a^2/c^2))=b/c
وبسبب وجود عدد لانهائي من ثلاثيات فيثاغورس فهناك عدد لانهائي من النقاط على دائرة الوحدة ذات الإحداثيات النسبية لكل من الفواصل والتراتيب

[gtdjpifvjkte]

Can a quadratic curve not have xy-terms?