Number theory, one of the branches of pure mathematics, is the study of numbers and their properties.
In many other branches of mathematics, numbers are used strictly
to convey information (e.g. 75%, 3 apples, 2`x`² - 1).
Number theory, on the other hand, studies numbers (usually
the integers or some subset thereof, like the natural
numbers) for their own sake.
The properties of numbers are indeed fascinating.

That's not to say that number theory has no practical value. Number theory is of great value in the information age. Knowledge of congruences and prime numbers has been invaluable in the areas of error-checking and -correcting codes and cryptography, respectively.

The first number theorist was probably
Pythagoras, who believed that
"all is number". The later Greeks, who
were no doubt influenced by him, valued number theory
(which they called *arithmetic*) above arithmetic
(which they called *logistic*).
The Greeks' mathematical discoveries were a cause (or perhaps a
symptom) of their contempt for practical mathematics.

Number theory is a fairly wide branch of mathematics. Here are some pages on Math Lair that discuss topics related to number theory:

- Congruences
- Digital Roots
- Divisibility Tests
- Factors and Multiples
- Prime Numbers
- Twin Primes
- The Goldbach Conjecture
- Fundamental Theorem of Arithmetic
- Perfect Numbers
- Square Numbers
- Triangular Numbers
- Diophantine Equations
- Pythagorean Triple Generator — source code
- Fermat's Last TheoremPell's Equation
- Niven Numbers