Larger Ideas:
- Rational and Integral Points on Circles and Ellipses
- Pythagorean Triples → Fermat’s Last Theorem
- Continued Fractions
It would be very cool to show that results from Number Theory and Abstract Algebra classes that can be adapted together to show results from the reading.
- Quadratic Reciprocity from 3240 and maybe simple group/ring/field theory from 3230 both are used for the rational and integral points on circles results.
For writing presentation
- Start with intro
- Then final result
- Based on how long those take, I can gauge how much time can be given to proving/showing lemmas and prior results.
Rational and Integral Points on Circles
Main Theorem: 12.1.10 Need:
- Lemma 12.1.9
- Lemma 10,3,4
- Theorem 12.1.5 (pretty big result, is a sum of two squares iff )
- Lemma 10.3.4
- Lemma 12.1.7 (algebra)
To do:
- Write up all prior theorems and lemmas
- Write up final result
- Write introduction
- Decide on necessary background information
- Decide on what motivation to include
- Finding rational points on curves is of interest in number theory
- We have a result that rational points exist on circles if and only if there exist integral points on them.
- This talk will focus on finding integral points on circles, which is an equivalent problem to finding when integers are sums of two squares.
- Decide on how much to include relating to the lemmas
Notes from Meeting:
- Condense “motivation” slides into one slide.
- “There are plenty of ”