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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 $c = a+1.$
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Diagram 2
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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}
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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 $(25-4)^2=21^2$, giving the Pythagorean triple $(20, 21,
29)$.
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Pythagorean Triples | $c-b=1^2$ ($p=q+1$) |
$c-b=3^2$ ($p=q+3$) |
$c-b=5^2$ ($p=q+5$) |
$c-b=7^2$ ($p=q+7$) |
$c-b=9^2$ ($p=q+9$) |
---|---|---|---|---|---|
$c-a=2 \times 1^2$
$(q=1)$
|
3, 4, 5 | 15, 8, 17 | 35, 12, 37 | 63, 16, 65 |
99, 20, 101
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$c-a=2 \times 2^2$
$(q=2)$
|
5, 12, 13 | 21, 20, 29 | 45, 28, 53 | 77, 36, 85 | 117, 44, 125 |
$c-a=2 \times 3^2$
$ (q=3)$
|
7, 24, 25 | 27, 36, 45 | 55, 48, 73 | 91, 60, 109 | 135, 72, 153 |
$c-a=2 \times 4^2$
$(q=4)$
|
9, 40, 41 | 33, 56, 65 | 65, 72, 97 | 105, 88, 137 | 153, 104, 185 |
$c-a=2 \times 5^2$
$(q=5)$
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11, 60, 61 | 39, 80, 89 | 75, 100, 125 | 119, 120, 169 | 171, 140, 221 |
$c-a=2 \times 6^2$
$(q=6)$
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13, 84, 85 | 45, 108, 117 | 85, 132, 157 | 133, 156, 205 | 189, 180, 261 |
$c-a=2 \times 7^2$
$ (q=7)$
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15, 112, 113 | 51, 140, 149 | 95, 168, 193 | 147, 196, 245 | 207, 224, 305 |
$c-a=2 \times 8^2$
$(q=8)$
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17, 144, 145 | 57, 176, 185 | 105, 208, 233 | 161, 240, 289 | 225, 272, 353 |
$c-a=2 \times 9^2$
$(q=9)$
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19, 180, 181 | 63, 216, 225 | 115, 252, 277 | 175, 288, 337 | 243, 324, 405 |