4383_Notes_Ch1

4383_Notes_Ch1 - Math 4383 Chapter 1 Page 1 of 9 Number...

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Math 4383 Chapter 1 Page 1 of 9 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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4383_Notes_Ch1 - Math 4383 Chapter 1 Page 1 of 9 Number...

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