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)