Study Guide State Fermat’s Last Theorem. When and where did Fermat live and work? What did he actually prove about his claim? Where did Fermat’s Last Theorem get its name from? What is the method of infinite descent? Who is credited with the proof of Fermat’s Last Theorem? When and where did this proof appear? What was his approach? What was Euler’s contribution to Fermat’s Last Theorem? When did he live and work? Who proved the case n=5? What was the contribution of Faltings? When was it made? What was the contribution and Frey and Ribbet? When was it made? What was the contribution of R. Taylor? When was it made? Show that in a primitive Pythagorean triple (d,e,f) d and e have to be of different parity Explain how to obtain a Pythagorean triple from similar plane numbers of the same parity. How can primitive Pythagorean triples be parameterized by pairs relatively prime numbers.

Sudy Guide What is Kummer’s contribution to Fermat’s Last Theorem? When did he live? What is an “ideal” number in the sense of Kummer. What is the definition of an irreducible number? What is the definition of a prime number in a number system such as Z[√-5] Give a number in Z[√-5] which does not have a unique factorization. Show that Z[w23] does not have unique factorization, where w23 is a 23rd root of unity. Let A=<2,1+ √-5> in Z[√-5]. Show that A·A =<4,2(1+√-5),-4+2√-5> =<2>. What was the contribution of Sophie Germain. When did she do her work? State the Theorem of Sophie Germain. What are the Case I and Case II solutions? What was the first step in Germain’s proof of her Theorem. Use Germain’s Theorem to show that there are no Case I solutions to the Fermat equation by showing that 3 is not a cube modulo 7 and that no two nonzero third power residues differ by 1. Where did Lamé go wrong? When did he make his claims? How long did they last? Who proved him wrong?