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primitive Pythagorean triples

Saturday 30 December 2017

Over the years, I’ve got the impression that students are as fascinated with Pythagorean triples as they are with prime numbers.  Many mathematical papers, books and websites delve into the various intriguing aspects of Pythagorean triples.  For example, see Dr Ron Knott’s impressive website Pythagorean Right-Angles Triangles (maybe some Exploration ideas to be found).  My interest in Pythagorean triples motivated me to make this a key component in the most recent additions to my HL and SL Problem-of-the-Day (there are now 200 in total !) – and also to construct a Geogebra applet for generating Pythagorean triples ... but, I’m getting a little ahead of myself.  First, here is some background information before presenting the Geogebra applet (but you can certainly jump ahead to the applet, if you wish).

A Pythagorean triple is a set of three positive integers $$\left( {a,\;b,\;c} \right)$$ satisfying the Pythagorean theorem $${a^2} + {b^2} = {c^2}$$.  Knowledge of (and, perhaps, fascination with) Pythagorean triples has existed for thousands of years.  The ancient Babylonian tablet known as Plimpton 322 contains a list of Pythagorean triples and is dated from around 1800 BCE.  The Great Pyramid of Giza in Egypt conforms almost precisely to the ratios in the 3-4-5 Pythagorean triangle.  A Pythagorean triangle is a triangle whose sides are a primitive Pythagorean triple.  A Pythagorean triple $$\left( {a,\;b,\;c} \right)$$ is “primitive” when the three numbers have no common factors (relatively prime, or coprime).  $$\left( {3,\;4,\;5} \right)$$ and $$\left( {5,\;12,\;13} \right)$$ are two well-known primitive Pythagorean triples.  $$\left( {6,\;8,\;10} \right)$$ is an example of a Pythagorean triple that is not primitive.  It’s obvious that there are an infinite number of Pythagorean triples because a multiple of any primitive Pythagorean triple is also a Pythagorean triple (just not primitive).  For example, Pythagorean triples $$\left( {6,\;8,\;10} \right)$$ and $$\left( {9,\;12,\;15} \right)$$ are multiples of $$\left( {3,\;4,\;5} \right)$$.  In his famous book, Elements, Euclid proved that there are an infinite number of primitive Pythagorean triples.

One of the interesting aspects of primitive Pythagorean triples is different methods for generating them.  The most well-known ‘formula’ for generating primitive Pythagorean triples appeared in Euclid’s Elements, and can be described as follows.

If m and n are two positive integers such that one is odd and the other is even, have no common factors other than 1 (relatively prime), and $$m > n$$, then the expressions $$a = {m^2} - {n^2},\;b = 2mn,\;c = {m^2} + {n^2}$$ will generate a primitive Pythagorean triple $$\left( {a,\;b,\;c} \right)$$.  This method can generate any possible primitive Pythagorean, but it’s not the only method.

A slightly different ‘formula’ is: if p and q are two relatively prime positive odd integers where $$p > q$$, then the expressions $$a = pq,\;b = \frac{{{p^2} - {q^2}}}{2},\;c = \frac{{{p^2} + {q^2}}}{2}$$ generate a primitive Pythagorean triple.  I prefer this ‘formula’ (or parameterization) and used it in composing the Geogebra applet below.  It is easier to check if both numbers are odd, and because a is the product of p and q it’s not difficult to construct a list of primitive Pythagorean triples where one of the ‘legs’ (a in this case) of the associated right triangle is the same; although only for values of a that are odd.  One just needs to choose an odd number and determine pairs of factors for the number.  For example, for the odd number 15 we have $$15 = 5 \times 3$$ and $$15 = 5 \times 1$$.  When $$p = 5$$ and $$q = 3$$, the primitive Pythagorean triple $$\left( {15,\;8,\;17} \right)$$ is generated; and when $$p = 15$$ and $$q = 1$$, the primitive Pythagorean triple $$\left( {15,\;112,\;113} \right)$$ is generated.  It’s a bit trickier with the hypotenuse (i.e. the value of c) because c is half of the sum of two squares (or 2c is sum of two squares).  However, it’s not too hard to show that 30 ($$2 \times 15$$) cannot be expressed as the sum of two squares.  Therefore, the only primitive Pythagorean triples in which the number 15 appears are $$\left( {15,\;8,\;17} \right)$$ and $$\left( {15,\;112,\;113} \right)$$.  A similar argument proves that $$\left( {9,\;40,\;41} \right)$$ is the only primitive Pythagorean triple in which the number 9 appears.

Use the Geogebra applet below – and have some fun generating primitive Pythagorean triples for various values of p and q.  What is true for all primitive Pythagorean triples generated when $$q = 1$$ ?   when $$p = q + 2$$ ?  when $$p = q + k,\;k \in {\mathbb{Z}^ + }$$ ?