# From Pythagoras to Fermat

Based on the name, if you are not familiar with the Pythagorean Theorem, check out the page The Infamous Pythagorean Theorem before moving on.

1. A Pythagorean Triple is a case where a, b, and c of the Pythagorean Theorem are positive integers (whole numbers). For c < 21. How many Pythagorean Triples are there?
2. How many Pythagorean Triples are there for c < 101?
3. How about for c < 1001?
4. Did you find a strategy that made problem 3 simpler than trying every combination of a, b, and c?
5. Although this research was originally inspired by right triangles, are there any Quadruples that satisfy the following? 6. Nothing says we have to stop at the normal Pythagorean Theorem. Are there any Triples for the following situations? 7. Fermat’s Last Theorem, which was not actually a theorem when it was proposed, has been the focus of many a mathematician’s work. Although it looks simple, pursuing a proof has demanded many advanced ideas and produced whole new fields of mathematics. Fermat wrote in the margins of a paper that there are no solutions to equation below for any n > 2. Was he correct? 8. Although you may not understand the proof of Fermat’s Last Theorem, dive into some of the writing and see if you can explain some of the basic principles of the proof and the mathematical fields that were created while pursuing it. Although Wikipedia is not a scholarly resource, its description of Fermat’s Last Theorem may be a good place to start your research.