Quadratic Forms

My next goal is to let you enter a primitive positive definite binary quadratic form with negative discriminant and you will get the equivalent reduced form. You can also get the steps to reduce the entered form. You can generalize these steps to provide a proof that a primitive positive definite binary quadratic form with negative discriminant is equivalent to a reduced form.

My final goal is to let you enter a prime p that is congruent to 1 modulo 4 and you will get the prime p expressed as a sum of squares of two integers. For example, if you enter the prime 61 then you will get 61 = 5*5 + 6*6. You can also get the steps to find these two integers whose squares sum up to the given prime p. You can generalize these steps to provide a proof of Fermat's result that every prime congruent to 1 modulo 4 can be expressed as the sum of two squares.

Here are some related links:

• imaginary quadratic fundamental discriminants of given class numbers
• Primes of the Form X2 + Ny2: Fermat, Class Field Theory,and Complex Multiplication

If you encounter problems, please email me.

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