Quadratic Form - History

History

The study of particular quadratic forms, in particular the question of whether a given integer can be the value of a quadratic form over the integers, dates back many centuries. One such case is Fermat's theorem on sums of two squares, which determines when an integer may be expressed in the form x2 + y2, where x, y are integers. This problem is related to the problem of finding Pythagorean triples, which appeared in the second millennium B.C.

In 628, the Indian mathematician Brahmagupta wrote Brahmasphutasiddhanta which includes, among many other things, a study of equations of the form x2 − ny2 = c. In particular he considered what is now called Pell's equation, x2 − ny2 = 1, and found a method for its solution. In Europe this problem was studied by Brouncker, Euler and Lagrange.

In 1801 Gauss published Disquisitiones Arithmeticae, a major portion of which was devoted to a complete theory of binary quadratic forms over the integers. Since then, the concept has been generalized, and the connections with quadratic number fields, the modular group, and other areas of mathematics have been further elucidated.