# Talk:Number theory

## The square of an even number is always even, and the square of an odd number is always odd

Try to prove it yourself. Hint: try squaring some smallish odd and even numbers, doing the calculation in binary (base 2) arithmetic. Dpbsmith 20:16, 18 April 2007 (EDT)

## Towards a better article

Lotsofsources to draw on, but this dicussionn is taken from http://www.math.uiuc.edu/ResearchAreas/numbertheory/guide.html

**Elementary Number Theory**
divisibility and prime factorization, residue classes, congruences, the quadratic reciprocity law, representation of numbers by forms, diophantine equations, continued fraction approximations and sieves.

**Analytic Number Theory**
An arithmetical phenomenon is represented by a related function, generally an analytic function of a complex variable. Information about the arithmetical problem is then extracted by analysis of the associated function.

**Algebraic Number Theory**
This deals with fields of algebraic numbers, that is with numbers which are roots of a polynomial equation with rational coefficients. Among areas of study are decomposition laws for primes in a number field, extensions of the famous quadratic reciprocity law and of Gauss and Jacobi sums, class numbers and units, and the connection between Stickelberger elements and p-adic L-functions.

**Probabilistic Number Theory**
In probabilistic number theory statistical limit theorems are established in problems involving "almost independent" random variables. Methods used include a combination of probabilistic, elementary and analytic ideas.

--AvengingAngel 18:36, 26 April 2007 (EDT)

The sentence, "Number theory typically deals with integers (as opposed to real numbers)" is awkward, because integers are not opposed to real numbers. Integers are a subset of real numbers.

- Excellent point, so I've clarified it to read, "Number theory typically deals only with integers (as opposed to all real numbers)."--Andy Schlafly 15:49, 27 December 2010 (EST)