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Pythagorean triples 2
Posted:
Jun 26, 2006 7:24 AM


A Pythagorean triple (PT) is a set of 3 integers {a,b,c} such that a^2+b^2c^2=0.
A PT can be represented as a rangled triangle. For any PT there is an infinite number of similar PTs, represented by similar rangled triangles.
Consider the set S of *nonsimilar* PTs. That is, none of the rangled triangles can be mapped onto any other one by enlargement.
It is straightforward to show that S is infinite.
Question 1:
Is S enumerable? If so, how?
Is there an algorithm that will produce (a) all nonsimilar PTs (b) only nonsimilar PTs (c) PTs only once?
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Because the PTs in S are nonsimilar, each PT can be specified by a single rational number, representing the proportion of the shorter sides of the PT. E.g. for the PT 5,12,13 this proportion is 5/12. Call this the PTspecific proportion
Question 2:
Can it be shown that, between any two PTspecific proportions (as defined above) p/q and r/s there is a third PTspecific proportion m/n, for a PT that is also a member of S, such that p/q<m/n<r/s? Alternatively, can it be shown that there are PTspecific proportions p/q and r/s such that there is no third PTspecific proportion lying between them?



