The most common Pythagorean Triple is (3, 4, 5).
Note that any natural number multiple of (3, 4, 5) is a Pythagorean triple. For example consider (2x3, 2x4, 2x5) which simplifies to (6, 8, 10).
A Pythagorean triple is a set of three natural numbers f, g, and h, such that f2 + g2 = h2.
Three numbers DO form a Pythagorean triple if a true result occurs when they are substituted into the Pythagorean Theorem formula.
Three numbers DO NOT form a Pythagorean triple if a false result occurs when they are substituted into the Pythagorean Theorem formula.
Usually a pythagorean triple (like the one above consisting of the natural numbers f, g and h is written as: (f, g, h).
Note also that any multiple of a Pythagorean triple is also a Pythagorean triple. For example if (f, g, h) is a Pythagorean triple, then (kf, kg, kh) [where k is any natural number] is also a Pythagorean triple.
The most common Pythagorean Triple is (3, 4, 5).
Note that any natural number multiple of (3, 4, 5) is a Pythagorean triple. For example consider (2x3, 2x4, 2x5) which simplifies to (6, 8, 10).
The numbers 4, 5 and 6 do NOT form a pythagorean triple as no permutation of the numbers when subtituted into the Pythagorean Theorem yield a true result.
When a Pythagorean triple consists of natural numbers which are relatively prime, it is more properly labeled a primitive Pythagorean triple.
From the first example, consider both pythagorean triples (3, 4, 5) and (6, 8, 10).
(3, 4, 5) IS a primitive Pythagorean triple since there is no natural number (other than 1) which divides evenly into all three numbers 3, 4 and 5.
(6, 8, 10) IS NOT a primitive Pythagorean triple since the natural number 2 divides evenly into all three numbers 3, 4 and 5.
Euclid's discovered a formula for generating Pythagorean triples. Substituting the two natural numbers m and n (with m > n) into the three equations below give the Pythagorean triple (f, g, h) such that f2 + g2 = h2.
f = 2mn g = m2 - n2 h = m2 + n2
NOTE: The Pythagorean triple generated by Euclid's formula is a primitive Pythagorean triple if and only if one of m and n are even and m and n are relatively prime.
Example One
Let m = 5 and n = 2
f = 2(5)(2) g = 52 - 22 h = 52 + 22
Simplifying results in f = 20, g = 21 and h = 29. Notice that the numbers 20, 21 and 29 are relatively prime. Notice also that with m = 5 and n = 2, m is odd and 2 is even (one of m and n are even) and 5 > 2 (m > n).
Substituting into the Pythagorean theorem results in the following:
Example Two
Let m = 5 and n = 3
f = 2(5)(3) g = 52 - 32 h = 52 + 32
Simplifying results in f = 30, g = 16 and h = 34. Notice that the numbers 30, 16 and 24 are NOT relatively prime, which means the the Pythagorean triple (30, 16, 34) is NOT a primitive one. Notice also that with m = 5 and n = 3, both m and n are odd (one of m and n are NOT even) and 5 > 3 (m > n).
Substituting into the Pythagorean theorem results in the following:
