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{{Main|Pythagorean triple}}
In ancient times it was known that a triangle whose sides were in the ratio 3:4:5 would have a right angle as one of its angles. This was used in construction and later in early geometry. It was also known to be one example of a general rule that any triangle where the length of two sides, each squared and then added together {{nowrap|1=(3<sup>2</sup> + 4<sup>2</sup> = 9 + 16 = 25)}}, equals the square of the length of the third side {{nowrap|1=(5<sup>2</sup> = 25)}}, would also be a right angle triangle.
This is now known as the [[Pythagorean theorem]], and a triple of numbers that meets this condition is called a [[Pythagorean triple]]; both are named after the ancient Greek [[Pythagoras]]. Examples include (3, 4, 5) and (5, 12, 13). There are infinitely many such triples,<ref name="Stillwell_2003">{{cite book | author = Stillwell J | year = 2003 | title = Elements of Number Theory | url = https://books.google.com/books?id=LiAlZO2ntKAC&pg=PA110 | publisher = Springer-Verlag | location = New York | isbn = 0-387-95587-9 | pages = 110–112 | access-date = 2016-03-17| authorlink=John Stillwell }}</ref> and methods for generating such triples have been studied in many cultures, beginning with the [[Babylonian mathematics|Babylonians]]{{sfn|Aczel|1996|pp=13–15|ps=}} and later [[Greek mathematics|ancient Greek]], [[Chinese mathematics|Chinese]], and [[Indian mathematics|Indian]] mathematicians.<ref name="auto"/> Mathematically, the definition of a Pythagorean triple is a set of three integers {{nowrap|(''a'', ''b'', ''c'')}} that satisfy the equation{{sfn|Stark|1978|pp=151–155|ps=}} {{nowrap|1=''a''<sup>2</sup> + ''b''<sup>2</sup> = ''c''<sup>2</sup>}}.
==== Diophantine equations ====
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