Pythagorean triple
A Pythagorean triple is a triple of positive integers, such that
. Pythagorean triples arise in geometry as the side-lengths of right triangles.
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Common Pythagorean Triples
These are some common Pythagorean triples: *=Primitive (see below)
For more pythagorean triples, see the Primitive Pythagorean Triple page.
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Pythagorean Triangles
Each positive integer solution of the diophantine equation defining the Pythagorean triples satisfies
. Thus, any triple of positive integers satisfying this equation also satisfies the triangle inequality, so the solutions correspond to right triangles with integral side lengths.
Primitive Pythagorean Triples
A Pythagorean triple is called primitive if its three members have no common divisors, so that they are relatively prime. Some triples listed above are primitive. Integral multiples of Pythagorean triples will also satisfy , but they will not form primitive triples. For example, all triples of integers of the form
, such as
, are Pythagorean triples.
General Form of Primitive Pythagorean Triples
Theorem. A triple of integers is a primitive Pythagorean triple if and only if it may be written in the form or
, where
are relatively prime positive integers of different parity.
Proof
Let be a primitive Pythagorean triple. If
and
both odd, then we must have
which is a contradiction, since 2 is not a square mod 4. Hence at least one of
and
, say
, is even. Then
must be odd, since
and
must be relatively prime. It follows that
is odd as well. It follows that the numbers
and
are positive integers. These positive integers must be relatively prime, since any common divisor of
and
must divide both
and
. Since
and
, it follows that
Since
must be an integer and
and
are relatively prime, it follows that
and
are perfect squares. Hence we may denote
and
for integers
and
. Since
is odd, it follows that
and
must have different parity, so
and
have different parity. Finally, we observe that
so any triple of the form specified in the theorem is a Pythagorean triple; it must furthermore be a primitive Pythagorean triple, since any common factor of
and
(both of which are odd integers, since
and
have different parity) must also be a factor of both
and
, which are integers with no common factor greater than 2.