Stinger #88

MATH STINGER #88
by Dr. Joel C. Fowler
Associate Professor of Mathematics

The puzzle for this issue concerns right triangles for which all three sides have integer lengths. The smallest right triangle with integer sides is a 3-4-5 right triangle. That is, the legs have lengths 3 and 4 while the hypotenuse has length 5. Note that the lengths of the sides increase by 1. If this triple is multiplied by 2 then the right triangle side lengths 6-8-10 are obtained. This has the sides increasing by 2. When multiplied by 3 the triple 9-12-15 is found which corresponds to a right triangle with sides that increase by 3. In fact, by multiplying 3-4-5 by an appropriate number, a right triangle can be found that has its sides increasing by any fixed amount. The puzzle for this issue is to determine if there are any right triangles, other than those obtained by scaling the 3-4-5 triangle, that have side lengths that increase by a fixed amount. If so, find them. If not, explain why not.