MATHmaniac aka CrUNchy

Diophantine Equations
The most generally enduring problem of number theory is probably that of diophantine equations. Greek mathematicians were quite adept at solving in integers x and y the equation
ux + by = c,
where a, b, and c are any given integers. The close relation with the greatest common divisor algorithm indicated the necessity of treating unique factorization as a primary tool in the solution of diophantine equations. The greek mathematicians gave some sporadic attention to forms of the more general equation

(1) f(x,y) = ax2 + bzy + cy2 + dx + ey + f = o,

but achieved no sweeping results. They probably did not know that every equation of this kind can be solved “completely” by characterizing a11 solutions in a finite number of steps, although they had success with special cases such as x2 -3y2 = 1. In fact, they used continued fraction techniques in both linear and quadratic problems, indicating at least esthetically a sense of unity. About 1750 euler and his contemporaries became aware this section presupposes some familiarity with elementary concepts of group, congruence, Euclidean algorithm, and quadratic reciprocity (which are reviewed in chapter 1).

Yet it was not until 1800 that Gauss gave in his famous disquisitiones arithmeticae the solution that still remains a mode1 of perfection. Now a very intimate connection developed between gauss’s solution and quadratic reciprocity, making unique factorization (in the linear case) and quadratic reciprocity (in the quadratic case) parallel tools. Finally, about 1896, Hilbert achieved the reorganization of the quadratic theory, making full use of this coincidence and thus completing the picture. Motivating problem in quadratic forms the first step in a general theory of quadratic diophantine equations was probably the famous theorem of Fermat relating to aquadratic form in x, y.

A prime number p is representable in an essentially unique manner by the form x2 + y2 for integral x and y if and only if p = 1 modula 4 (orp = 2). It is easily verified that
2 = l2 + 12, 5 = 22 + 12, 13 = 32 + 22, 17 = 42 + 12, 29 = 5~~+ 22, etc.,
whereas the primes 3, 7, 11, 19, etc., have no such representation. The proof of Fermat’s theorem is far from simple and is achieved later on as part of a larger result.

At the same time, fermat used an identity from antiquity:
(x2 + y2)(2’2 + y’2) = (xx’ -yy’) + (xy’ + x’y)2,
easily verifiable, since both sides equal x2x/2 + y~‘~ + x12y2 + z%‘~.
He used this formula to build up solutions to the equation
x2 + y2 = m
for values of m which are not necessarily prime. For example, from the results
3s + 2s = 13, (x = 3, y = 2),
2s + 1s = 5, (5’ = 2, y’ = l),
we obtain
72 + 42 = 65, (xx’ -yy’ = 4, xy’ + x’y = 7).

if we interpret the representation for 13 as

(-3)s + 2s = 13 (x = -3, y = 2)
whereas
2s + 1s = 5, (2’ = 2, y’ = l),
then we obtain
(-8)2 + l2 = 65, (xx’ -yy’ = -8, xy’ + x’y = 1);

Motivating problem in quadratic forms but it can verify that 65 = 72 + 42 = 82 + l2 are the only representations obtainable for 65 in the form x2 + y2, to within rearrangements of summands or changes of sign. if we allow the trivial additional operation of using (x, y), which are not relatively prime ((k~)~ + @y)~ = k2m), we can build up a11 solutions to (2), from those for prime m. Thus Fermat’s result, stated more compactly, is the following:
Let q(x, y> = x2 + y2.
then a11 relatively prime solutions (x, y) to the problem of representing qk y> = m
for m any integer are achieved by means of the successive application of two results called genus and composition theorems.

Genus theorem
(2) q@, y) = p
can be solved in integral x, y for p a prime of and only ~fp g 1 (mod 4), or p = 2. The representation is unique, except for obvious changes of sign or rearrangements of x and y.

Composition theorem
(3) q(x, y> qc%‘,y’> = q@x -yy’, s’y + ~y’>.
in the intervening years until about 1800, euler, lagrange, legendre, and others invented analogous results for a variety of quadratic forms. Gauss was the first one to see the larger problem and to achieve a complete generalization of the genus and composition theorems. The main result is too involved even to state here, but a slightly more difficult special result will give the reader an idea of what to expect.

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