davidlowryduda

davidlowryduda

davidlowryduda

davidlowryduda

davidlowryduda

davidlowryduda

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davidlowryduda

I wrote an article with Irving Kaplansky on indefinite binary quadratic forms, integral coefficients. At the time, I believe I used high-precision continued fractions or similar. It took me years to realize that the right way to solve Pell's equation, or find out the "minimum" of an indefinite form (and other small primitively represented values), or the period of its continued fraction, was the method of "reduced" forms in cycles/chains, due to Lagrange, Legendre, Gauss. It is also the cheapest way to find the class number and group multiplication for ideals in real quadratic fields, this probably due to Dirichlet. For imaginary quadratic fields, we have easier "reduced" positive forms.

A binary quadratic form, with integer coefficients, is some $$ f(x,y) = A x^2 + B xy + C y^2. $$
 The discriminant is $$ \Delta = B^2 - 4 A C. $$
 We will abbreviate this by $$ \langle A,B,C \rangle. $$
 It is primitive if $latex {\gcd(A,B,C)=1. }$ Standard fact, hard to discover but easy to check: $$ (A x^2 + B x y + C D y