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.
Read more about this topic: Quadratic Form
Famous quotes containing the word history:
“History is not what you thought. It is what you can remember. All other history defeats itself.
In Beverly Hills ... they don’t throw their garbage away. They make it into television shows.
Idealism is the despot of thought, just as politics is the despot of will.”
—Mikhail Bakunin (1814–1876)
“History has neither the venerableness of antiquity, nor the freshness of the modern. It does as if it would go to the beginning of things, which natural history might with reason assume to do; but consider the Universal History, and then tell us,—when did burdock and plantain sprout first?”
—Henry David Thoreau (1817–1862)
“The history of modern art is also the history of the progressive loss of art’s audience. Art has increasingly become the concern of the artist and the bafflement of the public.”
—Henry Geldzahler (1935–1994)