Pythagorean Triples
Pythagorean Triples
Tammie Strohl
MAT 126
David Gualco
November 9, 2009
Pythagorean Triples
Pythagorean Theorem states that the sum of the areas of the two squares formed along the two small sides of a right angled triangle equals the area of the square formed along the longest.

If a, b, and c are positive integers, they are together called Pythagorean Triples.
The smallest such Pythagorean Triple is 3, 4 and 5. It can be seen that 32 + 42 = 52 (9+16=25).
Here are some examples:

Endless
The set of Pythagorean Triples is endless.
It is easy to prove this with the help of the first Pythagorean triple, (3, 4, and 5):
Let n be any integer greater than 1: 3n, 4n and 5n would also be a set of Pythagorean triple. This is true because:
(3n)2 + (4n)2 = (5n)2


So, you can make infinite triples just using the (3,4,5) triple.
Euclid's Proof that there are Infinitely Many Pythagorean Triples
However, Euclid used a different reasoning to prove the set of Pythagorean triples is unending.
The proof was based on the fact that the difference of the squares of any two consecutive numbers is always an odd number.
For example, 22 - 12 = 4-1 = 3, 152 - 142 = 225-196 = 29.
And also every odd number can be expressed as a difference of the squares of two consecutive numbers. Have a look at this table as an example:

And there are an infinite number of odd numbers.
Since there are infinite number of odd numbers, and a part of them are perfect squares, there are an infinite number of odd squares (Since a fraction of infinity is also infinity). Therefore, there are infinite Pythagorean triples.
Properties
It can be observed that the Pythagorean triple consists of: all even numbers, or two odd numbers and an even number.
The Pythagorean triple can never be made up of all odd numbers or two even numbers and one odd number. This is true because:
(i) The square of an odd number is an odd number and the square of an even number is an
Tammie Strohl
MAT 126
David Gualco
November 9, 2009
Pythagorean Triples
Pythagorean Theorem states that the sum of the areas of the two squares formed along the two small sides of a right angled triangle equals the area of the square formed along the longest.

If a, b, and c are positive integers, they are together called Pythagorean Triples.
The smallest such Pythagorean Triple is 3, 4 and 5. It can be seen that 32 + 42 = 52 (9+16=25).
Here are some examples:

Endless
The set of Pythagorean Triples is endless.
It is easy to prove this with the help of the first Pythagorean triple, (3, 4, and 5):
Let n be any integer greater than 1: 3n, 4n and 5n would also be a set of Pythagorean triple. This is true because:
(3n)2 + (4n)2 = (5n)2


So, you can make infinite triples just using the (3,4,5) triple.
Euclid's Proof that there are Infinitely Many Pythagorean Triples
However, Euclid used a different reasoning to prove the set of Pythagorean triples is unending.
The proof was based on the fact that the difference of the squares of any two consecutive numbers is always an odd number.
For example, 22 - 12 = 4-1 = 3, 152 - 142 = 225-196 = 29.
And also every odd number can be expressed as a difference of the squares of two consecutive numbers. Have a look at this table as an example:

And there are an infinite number of odd numbers.
Since there are infinite number of odd numbers, and a part of them are perfect squares, there are an infinite number of odd squares (Since a fraction of infinity is also infinity). Therefore, there are infinite Pythagorean triples.
Properties
It can be observed that the Pythagorean triple consists of: all even numbers, or two odd numbers and an even number.
The Pythagorean triple can never be made up of all odd numbers or two even numbers and one odd number. This is true because:
(i) The square of an odd number is an odd number and the square of an even number is an