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PYTHAGOREAN TRIPLES

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 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 = (A2 - 1)/2

C = (A2 + 1)/2

Another equation derived by Plato was

(m2+1)2 = (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) = 1).

Enjoy these other topics in mathematics:

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