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Pythagorean triplet

Also called a Pythagorean triple, a set of three whole numbers that satisfies Pythagoras's theorem, i.e. the squares of two of the numbers add up to the square of the third number. Examples include (3, 4, 5), (5, 12, 13), and (7, 24, 25). These are called primitive triplets because they have no common divisors. If the members of a primitive triplet are multiplied by the same integer the result is a new (but not primitive) triplet.

In any primitive Pythagorean triplet, one, and only one, of the three numbers must be even (but can't equal 2); the other two numbers are relatively prime. There are infinitely many such triplets, and they are easy to generate using a classic formula, known since ancient times. If the numbers in the triplet are a, b, and c, then: a = n2 - m2, b = 2mn, c = m2 + n2, where m and n are two integers and m is less than n. Because the square root of two is irrational, there can't be any Pythagorean triplets (a, a, c). However, there are an infinite number of triplets (a, a + 1, c), the first three of which (apart from the trivial (0, 1, 1)) are (3, 4, 5), (20, 21, 29), and (119, 120, 169). There are also an infinite number of Pythagorean quartets (a, b, c, d) such that a2 + b2 + c2 = d2. This is simply the three-dimensional form of Pythagoras's theorem and can be interpreted as the fact that the point in three-dimensions with Cartesian coordinates (a, b, c) lies an integer distance d from the origin.

A formula that generates Pythagorean quartets is: a = m2, b = 2mn, c = 2n2; d = (m2 + 2n2) = a + c. Also note that b2 = 2ac. When m =1 and n = 1, we get the quarter (1, 2, 2, 3) – the simplest example. Although there are an infinite number of Pythagorean triplets, Fermat's last theorem, which is now know to be true, ensures that there are no triplets for higher powers.

Related entries

   • Euler's conjecture
   • multigrade

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