Math Help > Number Theory
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.
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
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.
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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