Math 4400/Number Theory/Fall 2012
Stuff to Know for the First Midterm
Definitions you need to know.
(1) What is a prime number?
(2) What is the gcd of two numbers?
(3) What are relatively prime numbers?
(4) What is the meaning of “a ≡ b (mod n)”?
(5) What is the definition of a group? A subgroup?
(6) What is the order of an element of a group? What is < g >?
(7) What is the order of a group, if it is finite?
(8) What are the groups ((Z/nZ), +) and ((Z/nZ)× , ·)?
(9) What is the Euler φ function?
How to’s.
(1) How to find the gcd of two numbers using Euclid’s algorithm.
(2) How to solve ax + by = c in integers using Euclid’s algorithm.
(3) How to find the inverse of a in (Z/nZ)× .
(4) How to find the order of a in (Z/nZ)× .
(5) How to “go backwards” in the Chinese Remainder Theorem.
(6) How to compute φ(n).
Theorems you should be able to state precisely.
(1) Uniqueness of factorization of natural numbers > 1.
(2) Lagrange’s Theorem
(3) The Chinese Remainder Theorem
(4) Fermat’s Little Theorem
(5) Euler’s Theorem
Stuff you should be able to prove.
(1)
(2)
(3)
(4)
If p is a prime and p|ab, then p|a or p|b.
Fermat’s little theorem assuming Lagrange’s theorem.
The Chinese Remainder Theorem
Anything I asked you to prove in a homework assignment.
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