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Topic: Pythagorean triples 2
Replies: 2   Last Post: Jun 26, 2006 11:16 AM

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cuthbert

Posts: 10
Registered: 6/26/06
Pythagorean triples 2
Posted: Jun 26, 2006 7:24 AM
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A Pythagorean triple (PT) is a set of 3 integers {a,b,c} such that a^2+b^2-c^2=0.

A PT can be represented as a r-angled triangle. For any PT there is an infinite number of similar PTs, represented by similar r-angled triangles.

Consider the set S of *non-similar* PTs. That is, none of the r-angled 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 non-similar PTs (b) only non-similar PTs (c) PTs only once?

*******************

Because the PTs in S are non-similar, 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 PT-specific proportion

Question 2:

Can it be shown that, between any two PT-specific proportions (as defined above) p/q and r/s there is a third PT-specific 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 PT-specific proportions p/q and r/s such that there is no third PT-specific proportion lying between them?



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