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Books > Science & Mathematics > Mathematics > Combinatorics & graph theory
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Binary Quadratic Forms - Classical Theory and Modern Computations (Paperback, Softcover reprint of the original 1st ed. 1989)
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Binary Quadratic Forms - Classical Theory and Modern Computations (Paperback, Softcover reprint of the original 1st ed. 1989)
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The first coherent exposition of the theory of binary quadratic
forms was given by Gauss in the Disqnisitiones Arithmeticae. During
the nine teenth century, as the theory of ideals and the rudiments
of algebraic number theory were developed, it became clear that
this theory of bi nary quadratic forms, so elementary and
computationally explicit, was indeed just a special case of a much
more elega, nt and abstract theory which, unfortunately, is not
computationally explicit. In recent years the original theory has
been laid aside. Gauss's proofs, which involved brute force
computations that can be done in what is essentially a two
dimensional vector space, have been dropped in favor of
n-dimensional arguments which prove the general theorems of
algebraic number the ory. In consequence, this elegant, yet
pleasantly simple, theory has been neglected even as some of its
results have become extremely useful in certain computations. I
find this neglect unfortunate, because binary quadratic forms have
two distinct attractions. First, the subject involves explicit
computa tion and many of the computer programs can be quite simple.
The use of computers in experimenting with examples is both
meaningful and enjoyable; one can actually discover interesting
results by com puting examples, noticing patterns in the "data,"
and then proving that the patterns result from the conclusion of
some provable theorem."
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