Pytha Essay

This essay has a total of 3356 words and 10 pages.

Pytha



Pythagorean Triples

Three integers a, b, and c that satisfy a2 b2 = c2 are called Pythagorean Triples. There
are infinitely many such numbers and there also exists a way to generate all the triples.
Let n and m be integers, n*m. Then define(*) a = n2 - m2, b = 2nm, c = n2 m2.

The three number a, b, and c always form a Pythagorean triple. The proof is simple: (n2 -
m2)2 (2mn)2 = n4 - 2n2m2 m4 4n2m2 = n4 2n2m2 m4 = (n2 m2)2. The formulas were
known to Euclid and used by Diophantus to obtain Pythagorean triples with special
properties. However, he never raised the question whether in this way one can obtain all
possible triples.The fact is that for m and n coprime of different parities, (*) yields
coprime numbers a, b, and c. Conversely, all coprime triples can indeed be obtained in
this manner. All others are multiples of coprime triples: ka, kb, kc.As an aside, those
who mastered the arithmetic of complex numbers might have noticed that (m in)2 = (n2 -
m2) i2mn. Which probably indicates that (*) has a source in trigonometry. But the proof
below only uses simple geometry and algebra.First of all, note that if a2 b2 = c2, then
(a/c)2 (b/c)2 = 1. With x = a/c and y = b/c we get x2 y2 = 1. This is the well known
equation of the unit circle with center at the origin. Finding Pythagorean triples is
therefore equivalent to locating rational points (i.e., points (x,y) for which both x and
y are rational) on the unit circle. For if (p/q)2 (r/s)2 = 1, multiplication by a common
denominator leads to an identity between integers.Rational numbers approximate irrational
to any degree of accuracy. Therefore, the set of rational pairs is dense in the whole
plane. So, perhaps, one might expect that any curve should contain a lot of rational pairs
or meander wildly to avoid them. But this is not the case. The recent proof of Fermat's
Last Theorem lets us claim that the curves xN yN = 1 with N*2 contain no rational
points. But there are simpler examples. From Lindemann's theorem, we conclude that the
graph of a perfectly smooth function y = ex contains a single rational point, (0,1).
Moreover, pulling the unit circle even a little aside may change the picture drastically.
Let (xk, yk) = ( 2/k, 3/k), and consider a unit circle with center at (xk, yk). As k
grows, the point approaches the origin, but for no k, such a circle contains a rational
point.Let t be defined by(1) t = y/(x 1).

Then t(x 1) = y andt2(x 1)2 = y2 = 1 - x2 = (1 x)(1 - x). We are not interested in
negative x. So let's cancel (1 x) on both sides. The result ist2(x 1) = (1 - x). Solving
for x we get(2) x = (1 - t2)/(1 t2)

From y = t(1 x) we also obtain(3) y = 2t/(1 t2)
Formula (1)-(3) show that t is rational iff both x and y are rational.There is another way
to look at the just described configuration.The configuration consists of the unit circle
centered at the origin and a straight line passing through the point (-1,0) which lies on
the circle. Unless the line is tangent to the circle, the two have a second common point.
In order to find this point, we have to solve simultaneously two equations: the quadratic
equation of the circle x2 y2 = 1 and the linear equation of the line. By eliminating
either x or y from the latter, and substituting the result into the former, we get a
quadratic equation in one variable with integer coefficients. One solution of this
equation is immediate - it is related to the point (-1,0), and is rational. Therefore, the
second solution of the equation is also rational and gives either x- or y-coordinate of
the second point of intersection.











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Continues for 5 more pages >>




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