## Pythagorean tripletAlso 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
= n^{2} - m^{2}, b = 2mn, c
= m^{2} + n^{2}, 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 a^{2} + b^{2} +
c^{2} = d^{2}. 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 = m^{2},
b = 2mn, c = 2n^{2}; d
= (m^{2} + 2n^{2}) = a + c.
Also note that b^{2} = 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 ## Related category• TYPES OF NUMBER | |||||

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