Pythagorean Triples

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.

Let's check it:

32 + 42 = 52

Calculating this becomes:

9 + 16 = 25

Yes, it is a Pythagorean Triple!

Triangles

Pythagoras

When a triangle's sides are a Pythagorean Triple it is a right angled triangle.

pythagoras squares a^2 + b^2 = c^2

See Pythagoras' Theorem for more details.

Example: The Pythagorean Triple of 3, 4 and 5 makes a Right Angled Triangle:

triangle 3 4 5

triangle 3 4 5 lego

 

Here are two more Pythagorean Triples:

5, 12, 13 9, 40, 41
   
5, 12, 13 Triangle 9, 40, 41 Triangle
52 + 122 = 132 92 + 402 = 412
25 + 144 = 169 (try it yourself)

 

And each triangle has a right angle!

List of the First Few

Here is a list of the first few Pythagorean Triples (not including "scaled up" versions mentioned below):

(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

Which also fits the formula a2 + b2 = c2:

62 + 82 = 102

36 + 64 = 100

If you want to know more about them read Pythagorean Triples - Advanced