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There are many different methods: the simpler methods will generate lots of triplets but not all. Comprehensive generators tend to be very complex.
So here is a simple one: Euclid's formula
Take any two positive integers x and y and suppose x > y. Then
A = x2 - y2
B = 2xy
C = x2 + y2
form a Pythagorean triplet.
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Are the numbers 9 12 15 pythagorean triples?
Yes
What set of numbers represents the lengths of the sides of a right triangle?
They are Pythagorean triples
Which set of numbers represent the sides of a right angle triangle?
They are Pythagorean triples
What combination of integers can be used to generate the pythagorean triple 8 15 17?
x=4 y=1
How does Euclid's Formula Work?
Euclid's Formula is a method of generating Pythagorean Triples. A Pythagorean Triple is a set of three positive integers (whole numbers), which satisfy the equation a2 + b2 = c2. The smallest Pythagorean Triple is 3, 4, 5. Euclid's Formula says this: If you choose two positive integers m and n, with m < n, then the three numbers n2 - m2, 2mn and n2 + m2 form a Pythagorean Triple. For example, if m = 5 and n = 7, n2 - m2 = 49 - 25 = 24, 2mn = 70, and n2 + m2 = 49 + 25 = 74. 24, 70, 74 is a PT, because 242 + 702 = 742. That's how to use Euclid's Formula. If the question means why does it work, then: (n2 - m2)2 + (2mn)2 = (n4 + m4 - 2n2m2) + (4m2n2) = n4 + m4 + 2n2m2, which is the same thing as (n2 + m2)2 . Two things to note are: The Formula does not generate all possible Triples, and it will generate Primitive Triples (ones with no common factor), only if m and n have no common factor, (except 1).
