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Published 1998 Revised 2009
Somewhat surprisingly every Pythagorean triple $( a, b, c)$,
where $a, b$ and $c$ are positive integers and $a^2+b^2 = c^2$, can
be illustrated by this diagram, in which the L shaped region has
area $b^2$, and the areas of the larger and smaller squares are
$c^2$ and $a^2$.
With this clue you can find some triples for yourself right
away. With an L strip of width 1 unit you get the whole class of
Pythagorean triples with $a$ and $c$ as consecutive integers, that
is $c = a+1.$

Diagram 2

This diagram (extended as far as required)
illustrates the fact that, for all $n$, the sum of the first $n$
odd numbers gives $n^2$. As we can see from the diagram to the
left:
\begin{eqnarray}1+3&=&2^2&=&4 \\ 1+3+5 &=&
3^2 &=& 9 \\ 1+3 +5 + 7 &=& 4^2 &=& 16 \\
1+3+5+7+9 &=& 5^2 &=& 25 \end{eqnarray}

To find other triples with L strips of different widths try
this for yourself:
For example, taking $p= 5$ and $q= 2$, the inner square has
dimensions $20$ by $20$ and the outer square has dimensions $29$ by
$29$, the width of the L strip is $9$, and the area of the L strip
is $(254)^2=21^2$, giving the Pythagorean triple $(20, 21,
29)$.

Pythagorean Triples  $cb=1^2$ ($p=q+1$) 
$cb=3^2$ ($p=q+3$) 
$cb=5^2$ ($p=q+5$) 
$cb=7^2$ ($p=q+7$) 
$cb=9^2$ ($p=q+9$) 

$ca=2 \times 1^2$
$(q=1)$

3, 4, 5  15, 8, 17  35, 12, 37  63, 16, 65 
99, 20, 101

$ca=2 \times 2^2$
$(q=2)$

5, 12, 13  21, 20, 29  45, 28, 53  77, 36, 85  117, 44, 125 
$ca=2 \times 3^2$
$ (q=3)$

7, 24, 25  27, 36, 45  55, 48, 73  91, 60, 109  135, 72, 153 
$ca=2 \times 4^2$
$(q=4)$

9, 40, 41  33, 56, 65  65, 72, 97  105, 88, 137  153, 104, 185 
$ca=2 \times 5^2$
$(q=5)$

11, 60, 61  39, 80, 89  75, 100, 125  119, 120, 169  171, 140, 221 
$ca=2 \times 6^2$
$(q=6)$

13, 84, 85  45, 108, 117  85, 132, 157  133, 156, 205  189, 180, 261 
$ca=2 \times 7^2$
$ (q=7)$

15, 112, 113  51, 140, 149  95, 168, 193  147, 196, 245  207, 224, 305 
$ca=2 \times 8^2$
$(q=8)$

17, 144, 145  57, 176, 185  105, 208, 233  161, 240, 289  225, 272, 353 
$ca=2 \times 9^2$
$(q=9)$

19, 180, 181  63, 216, 225  115, 252, 277  175, 288, 337  243, 324, 405 