Pythagorean Triples

 If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k.

A "Pythagorean Triple" is a set of positive integers, a, b and c that fits the rule:

a² + b² = c²

Example: The smallest Pythagorean Triple is 3, 4 and 5.

3² + 4² = 5²

9 + 16 = 25

Here is a list of the first few Pythagorean Triples:

(3,4,5)	(5,12,13)	(7,24,25)	(8,15,17)	(9,40,41)
(11,60,61)	(12,35,37)	(13,84,85)	(15,112,113)	(16,63,65)
(17,144,145)	(19,180,181)	(20,21,29)	(20,99,101)	(21,220,221)
(23,264,265)	(24,143,145)	(25,312,313)	(27,364,365)	(28,45,53)
(28,195,197)	(29,420,421)	(31,480,481)	(32,255,257)	(33,56,65)
(33,544,545)	(35,612,613)	(36,77,85)	(36,323,325)	(37,684,685)
... infinitely many more ...

Scale Them Up

The simplest way to create further Pythagorean Triples is to scale up a set of triples.

Example: scale 3,4,5 by 2 gives 6,8,10

However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides a = b = 1 and c = √2 is right, but (1, 1, √2) is not a Pythagorean triple because √2 is not an integer. Moreover, 1 and √2 do not have an integer common multiple because √2 is irrational.

Animation demonstrating the simplest case 

of the Pythagorean Triple: 3² + 4² = 5².

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, then 3n, 4n and 5n would also be a set of Pythagorean Triple. This is true because:

(3n)² + (4n)² = (5n)²

Examples:

n	(3n, 4n, 5n)
2	(6,8,10)
3	(9,12,15)
...	... etc ...

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.

Examples:

2² - 1² = 4-1 = 3 (an odd number),

15² - 14² = 225-196 = 29 (an odd number)

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:

n	n2	Difference
1	1
2	4	4-1 = 3
3	9	9-4 = 5
4	16	16-9 = 7
5	25	25-16 = 9
...	...	...

And there are an infinite number of odd numbers.

There is an infinite number of odd numbers. Since the perfect squares form a subset of the odd numbers, and a fraction of infinity is also infinity, it follows that there must also be an infinite number of odd squares. Therefore, there are an infinite number of Pythagorean Triples.

Properties

It can be observed that a Pythagorean Triple always consists of:

• all even numbers, or
• two odd numbers and an even number.

A 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 even number.
• (ii) The sum of two even numbers is an even number and the sum of an odd number and an even number is in odd number.

Constructing Pythagorean Triples

It is easy to construct sets of Pythagorean Triples.

When m and n are any two positive integers (m < n):

• a = n² - m²
• b = 2nm
• c = n² + m²

Then, a, b, and c form a Pythagorean Triple.

Example: m=1 and n=2

• a = 2² - 1² = 4 - 1 = 3
• b = 2 × 2 × 1 = 4
• c = 2² + 1² = 5

Thus, we obtain the first Pythagorean Triple (3,4,5).

Similarly, when m=2 and n=3 we get the next Pythagorean Triple (5,12,13).