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September 20th, 2010, 04:30 PM   #1
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Primitive Pythagorean Triples

Open question or not? We know that exist an infinitude of triples and primitive triples. And we know formulas to find all triples and formulas to find a infinitude of primitive triples (but not all). But is there any (parametric, simple) formula wich give us all and only primitive triples?
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September 20th, 2010, 04:51 PM   #2
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Re: Primitive Pythagorean Triples

To obtain all primitive Pythagorean triples use:

(m˛ - n˛, 2mn, m˛ + n˛)

where 0 < n < m, and (m,n) are co-prime and of opposing parity.
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September 20th, 2010, 06:17 PM   #3
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Re: Primitive Pythagorean Triples

Hm.. In fact I was thinking of something even more simples, a formula with no restrictions over the parameters.
What I mean is in that case seems like we are getting all the triples and then subtracting the ones which aren't primitives by restricting the parameters. Nothing wrong about that, but I was wondering if could be better in some way (for analysis, deductions or something) if we can get a formula where the variables are not restricted, where every pair of integer m,n give us a primitive triple. A more "natural" formula. With no gaps.
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September 20th, 2010, 08:12 PM   #4
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Re: Primitive Pythagorean Triples

From what I know, those restrictions on the integers (m,n) have to be lived with.

But, there are many here much more versed on number theory than I am, so perhaps, someone else knows a less restrictive method for getting all primitive triples.
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September 20th, 2010, 09:09 PM   #5
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Re: Primitive Pythagorean Triples

Your hopes are a bit unreasonable. You are looking for a particular class of solutions (namely primitive ones) which means you're going to have to have a divisibility condition on your parameters. If you just wanted all integer solutions, then those given above, without the conditions (other than the 0<n<m to make sure the solution makes sense and is non-trivial) would be exactly what you want. But if you don't want your conditions to deal with divisibility, you would have to introduce functions which implicitly dealt with the divisibility anyway (ie prime counting functions or some such). A significant part of solving equations over the integers (and even over the rational numbers) depends on the primes involved; you can't just wish them away. To a number theorist or an arithmetic geometer, the answer you have is as good as it gets.

That said, there are some cute methods (and not so off-the-beaten-path conceptually if you get in deep with solving these sorts of questions) for finding all of the triples. But in the end, they are just reinterpretations of the above formula.
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September 21st, 2010, 08:12 AM   #6
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Re: Primitive Pythagorean Triples

It is amazing, in some sense, that such a nice representation exists. Most problems don't have a solution nearly this nice! If it's not nice enough for you, I'm not sure what to say.
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September 21st, 2010, 09:30 AM   #7
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Re: Primitive Pythagorean Triples

I'm not saying that's not nice. I agree it is. ^^
Was just wondering about.. But I see the point. Thx guys.
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September 22nd, 2010, 05:24 PM   #8
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Re: Primitive Pythagorean Triples

As an aside, two consecutive terms of the Pell sequence will always generate a primitive triple, where the legs a and b always differ by 1.

The recursive definition is:

Pn+1 = 2Pn + Pn-1

where P1 = 1, P2 = 2

Here are the first few terms

1, 2, 5, 12, 29, 70, ...

Thus, the triple [(Pn+1)˛ - (Pn)˛, 2Pn+1Pn, (Pn+1)˛ + (Pn)˛] is always primitive.
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July 15th, 2011, 09:49 AM   #9
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Re: Primitive Pythagorean Triples

Quote:
Originally Posted by nonlinear
Open question or not? We know that exist an infinitude of triples and primitive triples. And we know formulas to find all triples and formulas to find a infinitude of primitive triples (but not all). But is there any (parametric, simple) formula wich give us all and only primitive triples?

The sequence of Fibonacci Boxes described here encodes the set of primitive triples, and provides a means of generating them all sequentially, without repetition or omission.

.

Definition:
A ‘Fibonacci Box’ is a 2x2 array containing four positive integers q', q, p, p' such that integers q,q' are coprime with q' odd, and obeying the Fibonacci Rule: q'+q = p and q+p = p' . Thus, q',p' are both odd, one of q,p is even, q < p , q' < p' and q', q, p, p' is a generalized Fibonacci sequence.

From any valid Fibonacci Box (the 'parent') we can obtain three more (three 'children') by applying the following template: Take the integers from the right-hand column of the parent and place them in three new Boxes as shown. (The values in black were re-computed using the Fibonacci Rule given above.)

|1 [color=#FF0000]1[/color]| |2 [color=#FF0000]1[/color]| |[color=#FF0000]3 1[/color]| |[color=#FF0000]1 3[/color]|
|2 [color=#FF0000] 3[/color]| |[color=#FF0000]3 [/color] 5| |4 7| |4 5|

An infinite sequence of Fibonacci Boxes can be constructed as follows. Beginning with the simplest Fibonacci Box on the left, place its 3 children to its immediate right. The next 3 Boxes in the sequence are the children of the second Box, produced in the same way, and so on.

|1 1| |2 1| |3 1| |1 3| |4 1| |5[color=#BFFFFF].[/color] 1| |1 5|
|2 3| |3 5| |4 7| |4 5| |5 1| |6 11| |6 7| .......

Each Box is unique. The Fibonacci rule guarantees there are no repetitions, and all such Boxes are eventually produced.

Primitive Pythagorean Triples

Each Box corresponds to a primitive triple [a,b,c] with sides:
a = q'p', b = 2qp, c= pp' - qq'.
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July 17th, 2011, 02:47 PM   #10
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Re: Primitive Pythagorean Triples

(Posts cannot be edited after the first hour or so.)

Edit: Or 0 hours, if the poster has fewer than 10 edits. This is an anti-spam measure.

Edit: It may be 48 hours. MarkFL and I have agreed that the time allowance is O(1) and leave the determination of the precise coefficient as an exercise for the reader.
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