Number theory

Number Theory is the queen of mathematics. It deals with the different properties of numbers such as integers, real numbers, fractions and other pure mathematical concepts. It includes determining the properties of prime numbers, divisibility of numbers and completing large calculations in short amounts of time. For example, an N-Queens solution can be checked for incorrectness by labeling each position (or square) of the chessboard with a number from 1 to n*n starting in the top-left corner and moving from left-to-right and top-to-bottom (or another symmetrical version of this), summing the values of the positions of the queens, and checking that value against $$\frac{n(n+1)}{2}$$. If they do not match, then answer is incorrect. However, if they do match then each queen is on a unique row and column, but the queens may or may not be attacking one another diagonally.