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:
“I believe that history might be, and ought to be, taught in a new fashion so as to make the meaning of it as a process of evolution intelligible to the young.”
—Thomas Henry Huxley (1825–95)
“It would be naive to think that peace and justice can be achieved easily. No set of rules or study of history will automatically resolve the problems.... However, with faith and perseverance,... complex problems in the past have been resolved in our search for justice and peace. They can be resolved in the future, provided, of course, that we can think of five new ways to measure the height of a tall building by using a barometer.”
—Jimmy Carter (James Earl Carter, Jr.)
“Anything in history or nature that can be described as changing steadily can be seen as heading toward catastrophe.”
—Susan Sontag (b. 1933)