Number Theory: Notes
January 14, 2016
1
Jan 11 16
Prop. 1. There are infinitely number of Primes.
Proof. Euclid
Suppose there are only infinitely many primes P1 , , Pt . Consider N = P1 Pt +
1. Suppose that we know N has some prime factor q. Then N = q k
Complexity of Arithmetic
1/8
Big-O Notation
Definition
We write f (n) = O(g (n) when there exists a constant C and
an integer M such that for n M,
f (n) C g (n).
Example
I
I
Observe that for f (n) = n2 + 5n + 1, we have
f (n) = O(n2 ).
Observe that for f
Gaussian Integers
1 / 11
The Gaussian Integers
Definition
The Gaussian integers are comprised by the set
G = cfw_a + bi : a, b Z,
where i =
1.
2 / 11
Translation to the Gaussian Integer Context
As we consider an analogue for the Gaussian integers of the
F
This is pdfTeX, Version 3.1415926-2.5-1.40.14 (MiKTeX 2.9) (preloaded
format=pdflatex 2015.4.25) 14 JAN 2016 16:56
entering extended mode
*Notes.tex
("D:\UNCW\Number Theory\Notes.tex"
LaTeX2e <2011/06/27>
Babel <v3.8m> and hyphenation patterns for english