Study Guide

1
Study Guide

• State Fermats Last Theorem.
• When and where did Fermat live and work? What did
  he actually prove about his claim?
• Where did Fermats Last Theorem get its name
  from?
• What is the method of infinite descent?
• Who is credited with the proof of Fermats Last
  Theorem? When and where did this proof appear?
  What was his approach?
• What was Eulers contribution to Fermats Last
  Theorem? When did he live and work?
• Who proved the case n5?

• 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.

2
Sudy Guide

• What is Kummers contribution to Fermats 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 Zv-5
• Give a number in Zv-5 which does not have a
  unique factorization.
• Show that Zw23 does not have unique
  factorization, where w23 is a 23rd root of unity.
• Let Alt2,1 v-5gt in Zv-5. Show that AA
  lt4,2(1v-5),-42v-5gt lt2gt.

• 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 Germains proof of her
  Theorem.
• Use Germains 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?