number theory - exams
note: more information
will be added as the exam dates near.
midterm exam: wed oct 30 (in class)
content: this exam will cover most of the first 3 chapters of the notes - more precisely,
-
sections 1.1-1.3, 1.5, 2.1-2.3, theorem 2.5.1 from sec 2.5, and sections 3.1-3.3.
while not necessarily a comprehensive list of topics covered on the exam, here
are the main things i am thinking about asking you to do on the exam. in
terms of types of problems, i would expect the following:
- answer true/false questions
- precisely state definitions/theorems with names (fundamental theorem of
arithmetic, Lagrange's theorem, Fermat's little theorem, Euler's theorem, etc)
- some computational exercises (by hand, no calculators allowed)
- write one or two simple non-computational proofs
(possibly of a result that was already
proved in class, an exercise, or something new)
in terms of content, be prepared to:
- determine if something is a group/ring/field
- compute the gcd of two integers (detailing steps of the Euclidean algorithm)
- use the extended Euclidean algorithm to find a solution (or prove
none exist) to a linear Diophantine equation: ax + by = c (possibly
over an imaginary quadratic ring)
- factor elements of irreducible quadratic rings into irreducibles
(which includes determining if an element in a quadratic ring is irreducible)
- check that an element of a quadratic ring has two distinct factorizations
or show why two factorizations are equivalent
- be able to use/prove basic divisibility tests
- do calculations mod n, such as determining of squares/non-squares or inverses of numbers
- use modular arithmetic to show certain equations have no solutions,
or possibly no solutions of a certain form
to prepare:
here is a selection of
to help you prepare for the exam. i encourage you to review first (using note, homeworks, etc), and then try the practice problems on your own like a mock test, before using notes/asking for help on them. bring whatever questions you have to class mon oct 28, which will be a review day. relatedly, i will move my office hours that week from their usual time:
office hours for week of oct 28: mon and wed 10-11am
final exam: mon dec 9, 1:30-3:30pm, phsc 1105
the final exam will be cumulative, covering up to section 4.5 of the notes. expect the types of problems i indicated above for the midterm.
in terms of content, here are the main things you should be comforable with
(this is not necessarily a comprehensive list of topics to study):
- all topics listed above for the midterms
- know what are subgroups and cosets, and Lagrange's theorem
- know Fermat's little theorem, and more generally Euler's theorem, and how they can be used to find inverses mod n
- exponentiate mod n using repeated squaring
- understand how RSA works, and be able to explain/carry out certain aspects of it (e.g., decryption, or generating a public key from a private key)
- fermat's two square theorem, and aspects of the proof
(wilson's theorem, lagrange's lemma, and the use of unique factorization in Z[i])
- understand the statement of the chinese remainder theorem and how to
use it to determine the euler phi function, and what is say about lifting
congruences mod m and mod n to congruences mod mn
- be able to use the law quadratic reciprocity (including the
first supplementary law) to determine whether a number is square mod a prime, or
apply to diophantine equations such as x^2 + dy^2 = n. (i will provide you with statements of law of quadratic reciprocity, and the first supplementary law.)
to prepare:
here is a selection of
try to do these on your own before the final day of class, and bring any questions you have then, or to office hours. if you would like to schedule a meeting outside of normal office hours, please let me know.
course home