A Pythagorean triple is defined as a
set of three positive integers (a,b,c) where a < b < c, such that
a2 + b2 = c2.
The sides of a right
triangle follows the Pythagorean Theorem,
a2 + b2 = c2
where a and b are the lengths of the legs of the right triangle
while c is the length of the hypothenuse.
Each Pythagorean triple forms the length of the sides
of a right triangle, whose perimeter is P = a + b + c.
A right triangle with sides of lengths 3, 4 and
5 is a special right triangle in that all the sides have whole number
lengths. The three numbers 3, 4 and 5 forms a Pythagorean triplet or
A Pythagorean triplet is a set of three
where the sum of the
squares of the first two is equal to the square of the third number.
Below are examples of Pythagorean triplets:
One equation satisfying a Pythagorean Triplet
A, B, C is
Given A is
B = (A2 - 1)/2
C = (A2 + 1)/2
Another equation derived by Plato was
= (m2-1)2 + (2m)2
where m is a natural number. The above equation
is called Plato's Formula.
Euclid has also another method, namely:
Given integers x and y,
A = x2 - y2
B = 2xy
C = x2 + y2
Pythagorean triples are called primitive triples if a,b,c are coprime, that is, if their pairwise greatest common divisors gcd(a,b) = gcd(a,c) = gcd(b,c) = 1. Because of their relationship through the Pythagorean theorem, a, b, and c are coprime if a and b are coprime (gcd(a,b)
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