Download Number Theory * Introduction (1/22)

Number Theory – Introduction (1/22)
• Very general question: What is mathematics?
• Possible answer:
The search for structure and patterns in the universe.
• Question: What is Number Theory?
• Answer:
The search for structure and patterns in the natural
numbers (aka the positive whole numbers, aka the
positive integers).
• Note: In general, in this course, when we say “number”,
we mean natural number (as opposed to rational number,
real number, complex number, etc.).
Some Sample Problems
in Number Theory
• Can any number be written a sum of square numbers?
• Can any number be written as sum of just 2 square
numbers? Experiment! See any patterns?
• Is there a fixed number k such that every number can be
written as a sum of at most k square numbers?
• Same question as the last for cubes, quartics
(i.e., 4th powers), etc.
• This general problem is called the Waring Problem.
More problems....
• Are there any (non-trivial) solutions in natural numbers to
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the equation a2 + b2 = c2? If so, are there only finitely
many, or are the infinitely many?
Are there any (non-trivial) solutions in natural numbers to
the equation a3 + b3 = c3? If so, are there only finitely
many, or are the infinitely many?
For any k > 2, are there any (non-trivial) solutions in
natural numbers to the equation ak + bk = ck? If so, are
there only finitely many, or are the infinitely many?
This last problem is called Fermat’s Last Theorem.
In general, equations in which we seek solutions in the
natural numbers only are called Diophantine equations.
And yet more... Primes!
• Definition. A natural number > 1 is called prime if....
• Are there infinitely many primes?
• About how many primes are there below a given number
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n? (The answer is called the Prime Number Theorem.)
Definition. Two primes are called twins if they differ by 2.
(Examples?)
Are there infinitely many twin primes? (This is, of course,
called the Twin Primes Problem.)
Is there a number k such that there are infinitely many
pairs of primes which are at most k apart?
The existence of such a number k was proved this past
summer!!!
More with primes
• Definition. We say two numbers a and b are congruent
modulo m if m divides b – a.
• Are there infinitely many primes which are congruent to 1
modulo 4? To 2 modulo 4? To 3 modulo 4?
• Can every even number be written as a sum of two
primes? (This is called the Goldbach Problem.)
• Can every odd number be written as a sum of three
primes? (This – sort of - is called Vinogradov’s Theorem.)
Assignment for Friday
• Obtain the text.
• Read the Introduction and Chapter 1.
• In Chapter 1 try out Exercises 1, 2, 3 and 5.