Pythagorean Triples |
Pythagorean Triples
Pythagoras's Theorem
For any right-angled triangle, The square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
a2 + b2 = c2
Pythagorean triples
- If two of the sides of a triangle have integer values, and using Pythagoras's theorem, the resulting third side is also has an integer value, then the lengths of the three sides form a Pythagorean triple
- The triple consist of three positive integers a, b and c such that a2 + b2 = c2
Examples
(3,4,5) | 32 + 42 = 9 + 16 = 25, 52 = 25 |
(5,12,13) | 52 + 122 = 25 + 144 = 169, 132 = 169 |
(6,8,10) | 62 + 82 = 36 + 64 = 100, 102 = 100 |
Primitive Pythagorean triples
A primitive triple is one for which the greatest common divisor is one. For example (3,4,5) is a primitive triple but (6,8,10) is not.
>There are methods of generating primitive triples.
Example
- Select any odd number a
- ((a2 + 1)/2)2 - ((a2 - 1)/2)2 = a2
- a, b = (a2-1)/2 and c = (a2+1)/2 form a primitive triple
- b and c form a pair with c = b + 1
- As b and c differ by one, then they cannot share a common factor, so the triple is primitive