Transcript Sample Questions - Purdue University

Study Guide
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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?
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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
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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?
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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>.