Counting sort initialization of the output array, each take time. Therefore, the time for the whole algorithm is the sum of the times for these steps, . Because it uses arrays of length and , the total space usage of the algorithm is also . For problem instances in which the maximum key value is significantly smaller than the number of items, counting sort can be highly space-efficient, as the only storage it uses other than its input and output arrays is the Count array which uses space . If each item to be sorted is itself an integer, and used as
Quadratic integer Pell's equations, and other questions related to integral quadratic forms. The study of rings of quadratic integers is basic for many questions of algebraic number theory. Medieval Indian mathematicians had already discovered a multiplication of quadratic integers of the same , which allowed them to solve some cases of Pell's equation. The characterization of the quadratic integers was first given by Richard Dedekind in 1871. A quadratic integer is an algebraic integer of degree two. More explicitly, it is a complex number formula_1, which solves an equation of the form , with "B" and "C" integers. Each quadratic integer that