Pythagorean Triple

A Pythagorean triple consists of three positive Integers ''a'', ''b'', and ''c'', such that ''a''² + ''b''² = ''c''². Such a triple is commonly written (''a'', ''b'', ''c''), and a well-known example is (3, 4, 5). If (''a'', ''b'', ''c'') is a Pythagorean triple, then so is (''ka'', ''kb'', ''kc'') for any positive integer ''k''. A '''primitive Pythagorean triple''' is one in which ''a'', ''b'' and ''c'' are Coprime .

The name is derived from the Pythagorean Theorem , of which every Pythagorean triple is a solution. The converse is not true. For instance, the Triangle with sides ''a'' = ''b'' = 1 and ''c'' = √2 is right, but (1, 1, √2) is not a Pythagorean triple because √2 is not an integer. Moreover, 1 and √2 don't have an integer common multiple because √2 is Irrational .

There are 16 primitive Pythagorean triples with ''c'' ≤ 100:

GENERATING PYTHAGOREAN TRIPLES

An effective way to generate Pythagorean triples is based on the observation that if ''m'' and ''n'' are two positive integers with ''m'' > ''n'', then

:

a = m^2 - n^2,\,

b= 2mn,\,

c = m^2 + n^2,\,

is a Pythagorean triple. It is primitive if and only if ''m'' and ''n'' are Coprime and one of them is even (if both ''n'' and ''m'' are odd, then ''a'', ''b'', and ''c'' will be even, and so the Pythagorean triple will not be primitive). Not every Pythagorean triple can be generated in this way, but every ''primitive'' triple (possibly after exchanging ''a'' and ''b'') arises in this fashion from a unique pair of coprime numbers ''m'' > ''n''. This shows that there are infinitely many primitive Pythagorean triples. This formula was given by Euclid (c. 300 B.C.) in his book ''Elements'' and is referred to as Euclid's formula.

An alternate form of the Euclid formula eliminates the negative sign by making use of the relation ''m'' = ''p'' + ''q'' and ''n'' = ''p'':

:

a = q(2p + q) \,

b = 2p(p+q) \,

c = (p + q)^2  + p^2 \,

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Pythagorean Triple