        Math & Science Home | Proficiency Tests | Mathematical Thinking in Physics | Aeronauts 2000

 CONTENTS Introduction Fermi's Piano Tuner Problem How Old is Old? If the Terrestrial Poles were to Melt... Sunlight Exerts Pressure Falling Eastward What if an Asteroid Hit the Earth Using a Jeep to Estimate the Energy in Gasoline How do Police Radars really work? How "Fast" is the Speed of Light? How Long is a Light Year? How Big is a Trillion? "Seeing" the Earth, Moon, and Sun to Scale Of Stars and Drops of Water If I Were to Build a Model of the Cosmos... A Number Trick Designing a High Altitude Balloon Pressure in the Vicinity of a Lunar Astronaut Space Suit due to Outgassing of Coolant Water Calendar Calculations Telling Time by the Stars - Sidereal Time Fields, an Heuristic Approach The Irrationality of The Irrationality of The Number (i)i Estimating the Temperature of a Flat Plate in Low Earth Orbit Proving that (p)1/n is Irrational when p is a Prime and n>1 The Transcendentality of Ideal Gases under Constant Volume, Constant Pressure, Constant Temperature and Adiabatic Conditions Maxwell's Equations: The Vector and Scalar Potentials A Possible Scalar Term Describing Energy Density in the Gravitational Field A Proposed Relativistic, Thermodynamic Four-Vector Motivational Argument for the Expression-eix=cosx+isinx Another Motivational Argument for the Expression-eix=cosx+isinx Calculating the Energy from Sunlight over a 12 hour period Calculating the Energy from Sunlight over actual full day Perfect Numbers-A Case Study Gravitation Inside a Uniform Hollow Sphere Further note on Gravitation Inside a Uniform Hollow Sphere Pythagorean Triples Black Holes and Point Set Topology Additional Notes on Black Holes and Point Set Topology Field Equations and Equations of Motion (General Relativity) The observer in modern physics A Note on the Centrifugal and Coriolis Accelerations as Pseudo Accelerations - PDF File On Expansion of the Universe - PDF File

##### Pythagorean Triples

Almost everyone knows of the "3-4-5 triangle," one of the right triangles found in every draftsman's toolkit (along with the 45-45-90). This triangle is different from most right triangles because it has three integer edges. Pythagoras' theorem tells us that the squares of the sides of a right triangle sum to give to the square of the hypotenuse:

32 + 42 = 52

I am often asked whether this relationship is unique, or if there are other right triangles with three integer edges as well.

When we randomly select two integers and add their squares, we usually acquire a non-integer square as a result; thus 32 + 52 = 34, or 42 + 72 = 65, and so on. Neither 34 nor 65 are integer squares. This type of result seems to be the general go of things; so the question posed is not without merit.
It turns out that there are an infinite number of right triangles with integer edges, which is relatively simple to see.
Consider a right triangle with edges a, b, and c such that

a2 + b2 = c2

The terms a and b are the sides of the right triangle so that a < c and b < c. Thus, we can subtract either a2 or b2 from both sides of the equation. Let's choose b2:

a2 = c2 - b2

Next, from basic algebra we see that we can factor the right-hand side:

a2 = (c + b)(c - b)

Now let's assume that a, b, and c are integers. Then a2, (c + b), and (c - b) must also be integers. Furthermore, since the product (c + b)(c - b) is equal to an integer square, both (c + b), and (c - b) must be integer squares. Let u and v be integers and set

c + b = 2u2
c - b = 2v2

where u2 > v2. (Why do we place a 2 in front of u2 and v2? Keep going-the reason should become clear shortly.) Then a2 = 4u2v2 = (2uv)2, and a = ± 2uv. Since a is a length, we can, without loss of generality, choose the + sign and set a = + 2uv.
Next, we can solve for b and c, obtaining

2b = 2(u2 - v2) and 2c = 2(u2 + v2)
or
b = u2 - v2
c = u2 + v2

(Now do you see the reason for the 2's? Try the same calculation again without them and evaluate the results as we do below!) With these two relations, and a = 2uv, we can set out to discover as many new right triangles with integer edges as we please.

The set of numbers, {a, b, c}, is called a Pythagorean triple. Here are a few examples to start you off. You can find as many more as you please.

 u v a b c 2 1 4 3 5 3 1 6 8 10 3 2 12 5 13 4 1 8 15 17 4 2 16 12 20 4 3 24 7 25