Binary quadratic form: Difference between revisions

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=== The representation problem ===
 
The oldest problem in the theory of binary quadratic forms is the '''representation problem''': describe the representations of a given number <math>n</math> by a given quadratic form ''f''. "Describe" can mean various things: give an algorithm to generate all representations, a closed formula for the number of respresentationsrepresentations, or even just determine whether any representations exist.
 
The examples above discuss the representation problem for the numbers 3 and 65 by the form <math>x^2 + y^2</math> and for the number 1 by the form <math>x^2 - 2y^2</math>. We see that 65 is represented by <math>x^2 + y^2</math> in sixteen different ways, while 1 is represented by <math>x^2 - 2y^2</math> in infinitely many ways and