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Number theory is the branch of mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study.  One of the key properties of interest to number theorists include factors of numbers (including common factors of two numbers), which leads to questions involving prime numbers and "modulo arithmetic".  This, in turn, leads to questions involving whether a number can be expressed as the square of another number (mod p), where p is prime.  And on and on.  This section briefly touches on some of these topics of number theory.

### Contents of this section: "Number Theory" gives you some basic definitions involving modulo arithmetic, GCD, etc.

Prime starts you off with the basics of prime numbers.  This section has a demonstration of the Sieve of Eratosthenes, an explanation of Prime Factorization and the Fundamental Theorem of Arithmetic, GCD, LCM and a little something about Counting -- how many subsets of a set exist? and how can you know this is an integer?

Perfect Squares discusses some facts about squares and square roots, including the fact that sqrt(n) is irrational unless n is a perfect square.

Highly Composite Numbers discusses an interesting pattern.  In a puzzle website, a "highly composite number" is one that has more factors than any smaller number.

Sum of SQRTs presents a puzzle which has an answer based in number theory

Cauchy-Schwarz inequality is (x1² + x2² + x3²)(y1² + y2² + y3²) ≥ (x1y1 + x2y2 + x3y3)², for reals xi and yj.  This is proved using the discriminant of a cleverly devised quadratic equation.

AM-GM Inequality is that the arithmetic mean of a set of positive numbers is always less than or equal to their geometric mean.  It is proved using Jensen's inequality.

Irrationality Proofs is a page that proves that e and π are irrational.

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### Related pages in this website

Pythagorean Triples

The webmaster and author of this Math Help site is Graeme McRae.