next up previous
Next: About this document ...

Pythagorean Triplets

ab

We seek to identify all positive integers a, b, and c so that a2+b2=c2. A triple (a,b,c) satisfying this equation is called a pythagorean triplet. Since any multiple of a pythagorean triplet is again one, we only seek primitive ones where a, b, and c have no common factor. By setting $x=\frac{a}{c}$ and $y=\frac{b}{c}$, we can first seek rational numbers x and y so that x2+y2=1. =2in
\epsfbox{pyth4.ps}
Now, if (x,y) is such a pair of rational numbers, the line from (x,y) to (-1,0) will have rational slope - call it t. Along this line, y=t(x+1), so x2 + t2(x+1)2=1 telling us that

(1+t2)x2 + 2t2 x = 1-t2

or

\begin{displaymath}
x^2 + \frac{2t^2}{1+t^2} x = \frac{1-t^2}{1+t^2}
\end{displaymath}

Complete the square on the above equation giving

\begin{eqnarray*}
(x + \frac{t^2}{1+t^2})^2 &= & \frac{1-t^2}{1+t^2}+\frac{t^4}{(1+t^2)^2}\\
& = & \frac{1}{(1+t^2)^2}\\
\end{eqnarray*}


Taking square roots gives

\begin{displaymath}
x=\frac{1-t^2}{1+t^2}.
\end{displaymath}

Write $t=\frac{u}{v}$ with u, v positive integers having no common factor. This gives

\begin{eqnarray*}
x &= & \frac{v^2-u^2}{v^2+u^2}\\
y &= & \frac{2uv}{v^2+u^2}\\
\end{eqnarray*}


Using $x=\frac{a}{c}$ and $y=\frac{b}{c}$ we see that

\begin{eqnarray*}
a &= &k(v^2-u^2) \\
b &= &k(2uv) \\
c &= &k(v^2+u^2) \\
\end{eqnarray*}


for some rational number k. By thinking about the highest power pr of an odd prime appearing in the denominator of k (written in lowest terms), one sees that pr divides uv, u2+v2, -u2+v2, 2u2, and 2v2. This means $p^{\frac{r}{2}}$ divides both u and v contradicting u and v having no common factor, UNLESS r=0. Similarly by noting exactly one of a and b can be odd, and choosing b to be the even number, we can eliminate powers of 2 in the denominator of k. This shows that k is a whole number and we obtain: Theorem: The pythagorean triplets with b even are exactly the triples of the form

\begin{eqnarray*}
a & =& v^2-u^2\\
b&=&2uv \\
c & =& v^2+u^2\\
\end{eqnarray*}


where u and v are positive whole numbers with no common factor.


next up previous
Next: About this document ...
root
2002-09-26