The Art and Craft of Problem Solving

7.4 DIOPHANTINE EQUATIONS 241

Here is a simple example of a problem with a "complete" solution, illustrating one

of the most importance tactics: factoring.

Example 7.4. 1 Find all right triangles with integer sides such that the area and perime­

ter are equal.

Solution: Let X,y be the legs and let z be the hypotenuse. Then z = J x^2 + y^2 by

the Pythagorean theorem. Equating area and perimeter yields

Basic algebraic strategy dictates that we eliminate the most obvious difficulties, which

in this case are the fraction and the radical. Multiply by 2, isolate the radical, and

square. This yields

or

�i - 4xy(x+y) +4(�+l+2xy) = 4(� +i).

After we collect like tenns, we have

�i -4xy(x+y) +8xy = O.

Clearly, we should divide out xy, as it is never equal to zero. We get

xy-4x-4y+8 = O.

So far, everything was straightforward algebra. Now we do something clever: add 8 to
both sides to make the left-hand side factor. We now have

(x -4)(y-4) = 8,

and since the variables are integers, there are only finitely many possibilities. The only

solutions (x,y) are (6, 8), (8,6), (5, 12), ( 12,5), which yield just two right triangles,

namely the 6-8- 10 and 5-12-13 triangles. _

The only tricky step was finding the factorization. But this wasn't really hard , as

it was clear that the original left-hand side "almost" factored. As long as you try to

factor, it usually won't be hard to find the proper algebraic steps.

The factor tactic is essential for finding solutions. Another essential tactic is to

"filter" the problem modulo n for a suitably chosen n. This tactic often helps to show

that no solutions are possible, or that all solutions must satisfy a certain fonn.^6 You

saw a bit of this already on page 230. Here is another example.

(^6) Use of the division algorithm is closely related to the factor tactic. See Example 5.4. 1 on page 165 for a nice
illustration of this.