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Pythagorean triplets - Essay Example

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Pythagorean triplets are integer solutions to the Pythagorean Theorem. Pythagoras lived around 500 BC, but Pythagorean triplets were first recorded as far back as 1700-1800 BC in Babylonia, inscribed on a clay tablet, known as Plimpton 322. A large number of integer pairs (a,c) for which there is an integer b satisfying Pythagoras' equation were systematically listed on it.
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Pythagorean Triplets Pythagorean triplets are integer solutions to the Pythagorean Theorem. Pythagoras lived around 500 BC, but Pythagorean triplets were first recorded as far back as 1700-1800 BC in Babylonia, inscribed on a clay tablet, known as Plimpton 322. A large number of integer pairs (a,c) for which there is an integer b satisfying Pythagoras' equation were systematically listed on it.
Let a, b, and c be positive real numbers. Then c is the length of the hypotenuse of a right triangle with side lengths a, b, and c if and only if;
a2 + b2 = c2
The Pythagorean triplet is an ordered triplet (a, b, c) of three positive integers such that
a2 + b2 = c2.
If a, b, and c are relatively prime, then the triple is called primitive.
For a right triangle, the 'c' side (the side opposite the right angle) is the hypotenuse. In general the 'a' side is the shorter of the two sides. There are some simple rules for determining a subset of Pythagorean triplets;
1. The 'a' side can take any odd number as its value.
2. The 'b' side is [(a2 - 1) / 2].
3. The 'c' side is equal to (b + 1).
OR
1. The 'a' side can take any even number as its value.
2. The 'b' side is equal to [(a/2)2 - 1].
3. The 'c' side is equal to (b+2).
Proof of above-mentioned rules:
Pythagorean theorem states a2 + b2 = c2 ------------------------- (Eq-1)
Case-I: Substituting c = b+1, in (Eq-1) we get,
a2 + b2 = (b + 1)2
a2 + b2 = b2 + 1 + 2b
a2 = 1 + 2b
a2 = 1+ 2 [(a2 - 1) / 2] (substituting value of b)
a2 = 1+ a2 -1
0 = 0 Hence proved
Some examples of such triplets are (1,0,1), (3,4,5), (5,12,13), (7,24,25), (9,40,41), (11,60,61), (13,84,85), etc.
Case-II: Substituting c = b+2, in (Eq-1) we get,
a2 + b2 = (b + 2)2
a2 + b2 = b2 + 4 + 4b
a2 = 4 + 4b
a2 = 4+ 4 [(a/2)2 - 1] (substituting value of b)
a2 = 4 + a2 -4
0 = 0 Hence proved
Some examples of such triplets are (2,0,2), (4,3,5), (6,8,10), (8,15,17), (10,24,26), (12,35,37), (14,48,50), etc.
From the above-mentioned discussion it is worth noting that the simplest triplets (1,0,1) and (2,0,2) are not triangles.
Resources:
1. Pythagorean Triplets, available online at http://www.friesian.com/pythag.htm.
2. Pythagorean Triples, available online at http://www.math.rutgers.edu/erowland/pythagoreantriples.html. Read More
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