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Math 4383
Chapter 1
Page
1
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Number Theory
Math 4383
Chapter 1 – Induction and the Binomial Coefficients
Number Theory – the study of the properties of the integers
More Active Definition – finding solutions to equations in the integers
Number theory is one of the oldest topics of mathematics, having roots that go back 2500 years
to the Greeks and even further to the Babylonians and others. As soon as we started counting, we
started noticing patterns in numbers.
Typical questions number theorists try to answer:
1.
How many primes are there?
2.
Characterize all positive integer triples x, y, z that are solutions to the equation
2
2
2
x
y
z
+
=
.
3.
Are there solutions to the equation
n
+
=
for n > 2?
4.
Find a formula (usable!) for the nth prime number.
5.
Are there infinitely many primes of the form
4
1
k
+
?
6.
Are there infinitely many primes of the form
2
1
+
?
7.
How can we decide (in a reasonable amount of time) whether a large integer is prime?
Question – can you tell which of the above questions have definite answers?
We will spend this semester looking at the basic definitions and functions studied in number
theory. We will use “elementary methods” to solve some questions. Elementary number theory
uses inductive reasoning, divisibility and factorization to answer questions about integers. These
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 Spring '09
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 Number Theory, Equations, Integers, Binomial

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