Pythagorean triples

A Pythagorean triple is any set of positive integers (a, b, c) which form the legs a, b, and the hypotenuse c of a right-angled triangle. It follows that
                        a2 + b2 = c2.

The best known example is (3, 4, 5), where
                        32 + 42 = 52.

A brief history

Despite their name, Pythagorean triples were known well over a thousand years before Pythagoras. The (3, 4, 5) triple, amongst others, was known to the ancient British builders of great stone monuments such as Stonehenge, and to the ancient Egyptians who used them in the building of the pyramids.

There is also a 3500 year old Babylonian clay tablet known as Plimpton 322, which shows the Babylonians had a good knowledge of Pythagorean triples. With four exceptions (which appear to have been careless errors), each row of the tablet actually lists a pair of numbers (b, c) which belong to a Pythagorean triplet.    























Pythagorean triples

A Pythagorean triple is any set of positive integers (a, b, c) which form the legs a, b, and the hypotenuse c of a right-angled triangle. So
                        a2 + b2 = c2.

The best known example is (3, 4, 5), where
                        32 + 42 = 52.
Exercise 1

Find some values of x for which Pythagoras’ formula produces Pythagorean triples. Can you find values of x which do not give such triples?

Example 2

Show that Plato’s formula gives rise to Pythagorean triples. Find a Pythagorean triple which is not given by the formula.

                             Check
Understanding the triples

It was, however, Pythagoras (6th Century BC) who first established a formula for generating the triples: for certain positive integers x, numbers a, b and c are given by

          a = x, b = 1/2(x2 – 1), c = 1/2(x2 + 1).

In the 4th Century BC, Plato contributed a more general formula:

              a = 2u,  b = u2 – 1,  c = u2 + 1,

for positive integers u.

Neither of these formulae generates all the Pythagorean triples.        
 
Ex 1: x = 1, (a, b, c) = (1, 0, 1), fails;
              x = 2, (a, b, c) = (2, 3/2, 5/2), fails;
        x = 3, (a, b, c) = (3, 4, 5); OK.

Ex 2: u = 1, (a, b, c) = (2, 0, 2), fails;
         u = 2, (a, b, c) = (4, 3, 5), OK;
           u = 3, (a, b, c) = (6, 8, 10), OK.






















Pythagorean triples

A Pythagorean triple is any set of positive integers (a, b, c) which form the legs a, b, and the hypotenuse c of a right-angled triangle. So
                        a2 + b2 = c2.

The best known example is (3, 4, 5), where
                        32 + 42 = 52.

Exercise 3

Use Euclid’s formula to find all Pythagorean triples for which c < 100. What happens when u and v are both even, or both odd?

Exercise 4

Find two Pythagorean triples for which c = 125.
                                Check
A general understanding

The problem of finding a general solution was finally solved by Euclid (3rd Century BC), with

                a = 2uv,  b = u2 – v2,  c = u2 + v2.

In this formula, u and v must be positive integers having no common factor, u > v, and u and v must have opposite parity: that is, if u is odd then v is even, and vice-versa.


Euclid

We find an echo of Euclid’s search for Pythagorean triples in the work of Diophantus (3rd Century AD) who in his Arithmetica Book II, Problem 8, seeks to divide a given square number into two squares.  

   

a = 2uv,  b = u2 – v2,  c = u2 + v2.
Ex 3
(u, v) = (2, 1); (a, b, c) = (4, 3, 5); 
(u, v) = (4, 1); (a, b, c) = (8, 15, 17);  
(u, v) = (6, 1); (a, b, c) = (12, 35, 37);
(u, v) = (8, 1); (a, b, c) = (16, 63, 65);
(u, v) = (3, 2); (a, b, c) = (12, 5, 13);  
(u, v) = (5, 2); (a, b, c) = (20, 21, 29);
(u, v) = (7, 2); (a, b, c) = (28, 45, 53);
(u, v) = (9, 2); (a, b, c) = (36, 77, 85);
(u, v) = (4, 3); (a, b, c) = (24, 7, 25);  
 (u, v) = (6, 3); (a, b, c) = (36, 27, 45);
(u, v) = (8, 3); (a, b, c) = (48, 55, 73);
(u, v) = (5, 4); (a, b, c) = (40, 9, 41);  
(u, v) = (7, 4); (a, b, c) = (56, 33, 65);
(u, v) = (9, 4); (a, b, c) = (72, 65, 97);
(u, v) = (6, 5); (a, b, c) = (60, 11, 61);
(u, v) = (8, 5); (a, b, c) = (80, 39, 89);
(u, v) = (7, 6); (a, b, c) = (84, 13, 85).

Ex 4  
         125 = 102 + 52 = 22 + 112      so (100, 75, 125) and (44, 117, 125).




















Fermat triples

It was in the margin of a French translation of Diophantus next to the above mentioned problem that Fermat (17th Century AD) writes his famous ‘last theorem’, namely

There are no positive integers a, b and c, and n > 2 such that  
an + bn = cn.

If we define a Fermat triple (a, b, c) to be a set of positive integers for which an + bn = cn for some integer n > 2, then Fermat’s last theorem is the equivalent of saying that Fermat triples do not exist. Much energy has since been expended in trying to prove just that.

 Useful links

This site gives a listing of Pythagorean Triples using the general formula

http://www.math.utah.edu/~alfeld/teaching/pt.html

Here is an interesting pictorial method of finding triples:

http://www.nrich.maths.org.uk/mathsf/journalf/may98/art1/

Further reading

An Introduction to the History of Mathematics, Eves, H., (Holt, Rinehart and Winston, 1969)

The Treasury of Mathematics, Midonick, H., Vol 2, (Pelican, 1968)

Megalithic Sites in Britain, Thom, A., (Oxford 1967)

Geometry and Algebra in Ancient Civilizations, van der Waerden, B.L. (Springer, 1983)