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Picturing Pythagorean Triples

Stage: 4 and 5
Article by Betty Hardy and Toni Beardon


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 of squares with notation
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}

Taking $a= 4$ and $c= 5$, the area of the outer ``L'' strip is $b^2= 9$ and this gives the first Pythagorean triple $4^2+ 3^2= 5^2$.

Before reading on, can you use more strips of unit width and extend the diagram to find Pythagorean triples with $b^2 = 25, 49, 81, 121$ etc.?

For every odd square number $b^2$, we can pick out a Pythagorean triple.

As the Diagram 2 shows, $c = a + 1$ and hence \begin{eqnarray} b^2 = c^2 - a^2 \\ b^2 = (a+1)^2 - a^2 \\ b^2 = a^2 + 2a +1 -a^2\\ = 2a + 1 \end{eqnarray} The next odd square after $3^2 = 9$ is $5^2 = 25$
Let $$25 = b^2$$
$$25 = 2*12 + 1$$
giving $a$ = 12 and from there the triple $5^2+12^2=13^2$

The next one in this class is $b=7$ giving $a=24 = \frac{(49-1)}{2}$ and $c=25$

Giving the triple $24^2+7^2=25^2$, and so on.

This method gives us infinitely many Pythagorean triples, but what about others where the width of the L strip (the difference between $a$ and $c$) is more than 1?
Diagram of square within square showing notation
To find other triples with L strips of different widths try this for yourself:

  • choose any two positive integers $p$ and $q$ that have no common factors and with $p > q$; \item calculate $2p q$ and $p^2+q^2$; \item draw two squares with sides $a=2p q$ and $c = p^2 + q^2$ as shown in Diagram 3.
  • The width of the L strip is $w = (p-q)^2$ and its area is $b^2 = (p^2 -q^2)^2$, and if you wish you can check this algebra 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 $(20, 21, 29)$.

The process given by these instructions produces what is called a parameterisation and $p$ and $q$ are called parameters. It is easy to show by simple algebra that, by choosing positive integers $p$ and $q$, with $p> q$, and taking
$$b = p^2-q^2$$
$$a = 2p q$$
$$c = p^2 + q^2$$
we get a Pythagorean triple.

This does not give all triples, for example it does not give ($9,12,15)$, but it does give all triples for which $a, b$ and $c$ have no common factor (`primitive' triples). Every other triple can then be found by calculating $( k a, k b, k c)$ for some integer $k$.

These results are proved in the article Pythagorean Triples II .

We shall concentrate on finding all the triples where $a, b$ and $c$ have no common factor. If $p$ and $q$ have a common factor, or if $p$ and $q$ are both odd, then $a, b$ and $c$ will have a common factor.

We only need the cases where one of $p$ and $q$ is even and the other is odd. In such cases the width of the L strip $= c-a=(p-q)^2$ is always an odd square.

The table below (extended in the obvious way) shows all the `primitive' Pythagorean triples, and some non-primitive triples as well.

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

$c-a=2 \times 2^2$
$(q=2)$
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
$c-a=2 \times 5^2$
$(q=5)$
11, 60, 61 39, 80, 89 75, 100, 125 119, 120, 169 171, 140, 221
$c-a=2 \times 6^2$
$(q=6)$
13, 84, 85 45, 108, 117 85, 132, 157 133, 156, 205 189, 180, 261
$c-a=2 \times 7^2$
$ (q=7)$
15, 112, 113 51, 140, 149 95, 168, 193 147, 196, 245 207, 224, 305
$c-a=2 \times 8^2$
$(q=8)$
17, 144, 145 57, 176, 185 105, 208, 233 161, 240, 289 225, 272, 353
$c-a=2 \times 9^2$
$(q=9)$
19, 180, 181 63, 216, 225 115, 252, 277 175, 288, 337 243, 324, 405

There is a dual illustration of each triple where the smaller square has side $p^2 - q^2$ and the width of the L strip is $2q^2$. In the dual case for $p=5$ and $q =2$ the inner square has dimensions $21$ by $21$ and the width of the L strip is $8$ units giving the triple $(20,21,29)$.

Given a Pythagorean triple $(a, b, c)$ you can always `work back' to find $p$ and $q$, and given $p$ and $q$ you can always calculate $a, b$ and $c$. You could compete with a friend, both choose a value for $p$ and $q$ and compute a Pythagorean triple $(a, b, c)$ which you give to the other person. Then without using calculators, see who can be the first to find the values of $p$ and $q$. You may need to keep $p$ and $q$ small at first to make this easy for yourselves but as you get better you can take bigger values of $p$ and $q$.