Triples and quadruples: from Pythagoras to Fermat
Pythagoras' theorem. Image: Wapkaplet.
If there’s one bit of maths you remember from school it’s probably Pythagoras’ theorem. For a right-angled triangle with sides , , , where is the side opposite the right angle, we have
If three positive whole numbers , and satisfy this equation — if they form the sides of a right-angled triangle — they are said to form a Pythagorean triple.
One question that intrigued Pythagoras himself, as well as other ancient Greek mathematicians, is how to generate Pythagorean triples. If I give you a positive number , can you find two numbers and so that , and form a Pythagorean triple? In this article we’ll explore this question, and we’ll also see how the idea extends to sets of four numbers, called Pythagorean quadruples.
First of all, here are some examples of Pythagorean triples:
Some Pythagorean triples.
The triples written in red are multiples of each other and so are the triples written in blue: you get and by multiplying the components of by 2, 3 and 4 respectively, and you get by multiplying the components of by 2.
In general, if is a positive whole number and is a Pythagorean triple, then so is , since
Geometrically, if one Pythagorean triple is a multiple of another, then the corresponding triangles are similar.
Pythagoras as depicted by Raffaello Sanzio in his painting The school of Athens.
If a Pythagorean triple isn’t a multiple of another Pythagorean triple, then we say that it is a primitive triple. You can recognise a primitive Pythagorean triple by the fact that the numbers and do not have a common divisor. In our example is a primitive Pythagorean triple while and are not. Similarly is a primitive triple whereas is not.If you're given a Pythagorean triple it's easy to generate new non-primitive ones simply by taking its multiples. But given just a number, can you find a Pythagorean triple with that number as one of its components? One method for doing this has been attributed to Pythagoras himself. First note that if
Now consider the two expressions
They differ by exactly so the two expressions
Plato (left) with Aristotle (right) as depicted by Raffaello Sanzio in his painting The school of Athens.
Therefore, if we choose
For the numbers , and to represent a Pythagorean triple we need
to be whole numbers. This means that both
need to be even, which in turn implies that needs to be odd. But the square of a number is odd only if the number itself is odd, so this method only works for odd .
There is however an easy way to derive a formula for even values from the above. If , and form a Pythagorean triple of the form described above, then so do
This method for generating triples from even numbers has been attributed to Plato. Here is a list of Pythagorean triples generated from both even and odd numbers using these two methods:
Since the methods give us a triple for every positive whole number we see that there are infinitely many Pythagorean triples. But can these methods generate all of them? The answer is no. For example, the triple is absent from the list above. A general formula was described by Euclid in his famous book The Elements. Take any two positive whole numbers and with . Similarly to our reasoning above, notice that
differ by So setting
Since and are positive whole numbers and all three numbers , and are also positive whole numbers, so we have a Pythagorean triple. Every primitive Pythagorean triple can be generated from a unique pair of numbers and , one of which is even. And once you have the primitive ones you can generate all Pythagorean triples simply by multiplying. So Euclid’s formula really does give you all the triples there are.
Now let’s look at Pythagorean quadruples which consist of four positive whole numbers instead of three. In a Pythagorean quadruple the sum of squares of first three numbers gives us the square of the fourth:
Some Pythagorean quadruples.
We can generate a Pythagorean quadruple from any two numbers and simply by noting that
and gives us a Pythagorean quadruple.
This also gives us a way of generating a Pythagorean quadruple from a single even number . Firstly, note that if is even, then is even. Now find two numbers and so that Set
gives us our Pythagorean quadruple. For example, if , then so choose and We get the quadruple with
For we have We now have two choices as and The first choice gives the quadruple with
The second choice gives the quadruple with
You can continue to generate quadruples from even numbers in this way.
Can we generate all Pythagorean quadruples?
Not all Pythagorean quadruples are of the form
so not all of them can be generated using the method we just described — we need to be a little cleverer. Suppose that you’re given two numbers and Now find a number which divides but so that If and are both even, then we also require that itself is even.
Euclid (the man with the compass) as depicted by Raffaello Sanzio in his painting The school of Athens.
But are , , and positive whole numbers? This is why we’ve imposed conditions on You can show that as long as and are either both even, or if one is even and one is odd, then the conditions ensure that , , and are positive whole numbers.
(Click here to see why.)
If and are both odd it is impossible to generate a Pythagorean quadruple from them by this method.
But the important point is that you can construct every primitive Pythagorean quadruple from two numbers and in the way we’ve just shown. And again, once you have the primitive ones, you can get all the others just by multiplying.
Generating a series of squares
Another nice thing to notice is that using our mechanism for generating triples, we can make sums of squares of any length. Let’s start with the triple We can generate another triple starting with the number 5: it’s Thus we have
Rearranging the second equation gives
Substituting this into the first equation and rearranging gives
so we have the quadruple Proceeding in a similar way, always using the biggest of the current set of numbers to generate a new triple, we can construct the quintuple and the sextuple and so on, ad infinitum.
Cubes and beyond
Pythagorean quadruples consist of a sum of squares, but what if we look at sums of cubes of the form
These are called cubic quadruples. Here are a few examples (again, quadruples written in red, blue or green are multiples of each other).
Some cubic quadruples.
We won’t explore how to generate them here, but instead ask a question that turns out to be more interesting: are there also cubic triples? This question is the subject of one of the most famous results in mathematics: Fermat’s last theorem. The theorem says that there are no three positive whole numbers , and such that
In fact, the theorem says more than that: for any positive whole number greater than two it is impossible to find three positive whole numbers , and such that
The result was made famous by the French mathematician Pierre de Fermat in 1637. Fermat wrote in the margin of his book that he had "discovered a truly marvelous proof of this, which this margin is too narrow to contain". For over 300 years mathematicians desperately tried to reconstruct this marvellous proof, but they didn't succeed. It was not until 1995 that the mathematician Andrew Wiles proved the result, using sophisticated mathematics Fermat could not possibly have known about.
You can read more about Fermat's last theorem on Plus.
About the author
Chandrahas Halai is a mathematics enthusiast from the land of the Shulba sutras, the Bakhshali manuscript, and mathematicians like Aryabhatt, Brahmagupta, Bhaskaracharya, Ramanujan and many more. He is a consultant in the field of computer aided engineering, engineering optimisation, computer science and operations research. He writes research papers, articles and books on mathematics, physics, engineering, computer science and operations research. In his spare time he likes doing nature photography and painting.