A Pythagorean triple is defined as a
set of three positive integers (a,b,c) where a < b < c, such that
a^{2} + b^{2} = c^{2}.
The sides of a right
triangle follows the Pythagorean Theorem,
a^{2} + b^{2} = c^{2}
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
Pythagorean triple. A Pythagorean triplet is a set of three
whole numbers
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:
3 
4 
5 
5 
12 
13 
7 
24 
25 
9 
40 
41 
11 
60 
61 






One equation satisfying a Pythagorean Triplet
A, B, C is Given A is
odd, then
B = (A^{2}  1)/2
C = (A^{2} + 1)/2
Another equation derived by Plato was
(m^{2}+1)^{2}
= (m^{2}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 = x^{2}  y^{2}
B = 2xy
C = x^{2} + y^{2}
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)
= 1).
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