Copyright ©2008 by Paul Niquette. All rights reserved..
ack in some indefinitely early age, we have been taught the Pythagoras Theorem, "The square of the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides." For more than 25 centuries, the Theorem has been learned and found useful. Never mind that in modern times it has been misquoted in a popular film and lampooned in a world-class groaner.
Just as famous in our scholastic youth was the "three|four|five triangle," wherein integers 3 and 4 provide the dimensions for the two sides and 5 represents the length of the hypotenuse. Accordingly, the square of five (52 = 25) equals the square of three (32 = 9) plus the square of four (42 = 16).
An unlimited number of triangles can each have 5 for their
hypoteneese. The square of their common hypotenuse 25 can be produced
as the sum of 12 integer pairs...
The members of only one pair, 9+16=25, are both squares
of integers. The dimensions in other pairs are generally not integers,
one or both will be irrational numbers formed as products, each
pair including the square root of at least one prime integer...
In mathematics, positive integers are also called natural numbers. For Natural Hypoteneese puzzle, we shall define a natural hypotenuse as an integer that can be formed as the square root of the sum of the squares of two integers. The "three-four-five triangle" offers the smallest natural hypotenuse. Don't you wonder how many other right triangles exist that have integers for all three sides?
epicted in the graph below are the dimensions of 15 triangles with Natural Hypoteneese plotted according to the lengths of their non-hypoteneese. The "three|four|five triangle" sits in the lower left corner of the graph. Some observations...
The triangle 5|12|13 is also the
member of another whole family, which begins with our old friend
"three|four|five triangle." This family of Natural
Hypoteneese includes 7|24|25 and 9|40|41,
which you will notice do not lie along a straight line in the graph.