Triangles With Integer Sides

  In his book on Diophantus' work Arithmetic or Numbers Heath states that Pythagoras gives a family of solutions which provide integer values for the sides of a right triangle. How could Pythagoras have accomplished this? The Pythagorean Theorem states that the sum of the squares of the sides are equal to the square of the hypotenuse or a²+b²=c². We can subtract the square of one side from the square of the hypotenuse and then solve for this side using the other side and an auxiliary variable x.

We get a rational expression for side a and can factor the numerator. Side a is presumed to be an integer so lets assume that b-x is divisible by 2x with the quotient being equal to another auxiliary variable λ. We can then express all the sides of the triangle in terms of  the variables x and λ. Noting that all three sides have a common factor, x, we can reduce the expressions for the sides by ignoring the larger similar triangles.

This is the family of solutions attributed to Pythagoras. Evaluating the formulas for integer values of λ does indeed give integer solutions. Note that b is always an odd number. We can find another set of solutions for which b is an even number by manipulating the formulas a little.

We can compare the two sets of numbers by plotting them.

Multiplying all three sides of these triangles by a common integer factor gives more integer solutions. We can also multiply by a rational number to rescale and find smaller similar triangles.

 That Pythagoras was able to arrive at the first set of solutions indicates that he had a fairly good grasp of algebra.