Maths Problem

[KrazyHorse]

Never mind previous post. I assumed something in "proof" of pythagorean question that's not strictly true. Now not sure if it's obvious how to classify all possible pythagorean triplets.

[KrazyHorse]

I of course generate an infinite series of answers but they don't exhaust possibilities. My inductive step was flawed...

[Snowflake]

The question is, for each number, does there exist a unique set of pythagorean triplets where this number is the smallest?

[KrazyHorse]

If n = 2 (mod4) then obviously not.

[Snowflake]

Can there be multiple sets for a number though?

[Snowflake]

Looks like the answer is yes.

Found this page.

[KrazyHorse]

Reasoning as follows:

y^2 - x^2 = n^2

Obviously either x odd and y odd or x even and y even (if not then n^2 must be odd, a contradiction)

If x even and y even then there is a smaller equivalent triplet

Therefore x odd and y odd. Let y = x+2i => y^2 -x^2 = 4ix + 4i^2 = 4(ix + i^2). However, for x odd ix + i^2 is even for all i (if i odd then you are adding an odd to an odd; if i even then an even to an even) so y^2 - x^2 is divisible by 8. However n = 2(mod 4) cannot give n^2 = 0(mod 8), so x odd and y odd impossible

[KrazyHorse]

Originally posted by Snowflake
The question is, for each number, does there exist a unique set of pythagorean triplets where this number is the smallest?

Ah. Now I understood what you mean. No, I've already admitted that there is not. We've already had a counterexample as well...

[Lul Thyme]

If I remember correctly there are two infinite families of irreducible integer pythagorean triple (meaning no common divisor to each...)
the
3,4,5
5,12,13
7,24,25
etc...
is one
let me check if I can find my reference

[Lul Thyme]

My reference is Number theory and its history by oystein ore.
I seemed to remember correctly, but Im a bit in a hurry, so Ill post the details later.

BTW Snowflake, keep em coming

[civman2000]

Originally posted by KrazyHorse
In fact, it's quite easy to show that up to a scale factor all pythagorean triplets are equivalent to a = (2n + 1), b = (2n^2 + 2n), c = (2n^2 + 2n + 1)

This may be true (I would conjecture it is; it wouldn't take long to check but I'm feeling lazy) but this:

It is the case up to a scale factor, as I've said. For all intents and purposes 3 4 5, 5 12 13, 7 24 25 etc. are the only pythagorean triplets. All others are direct multiples of these.

...is not. Take, say, 20-21-29. To get c-b=1 (where b is the even one), you would have to divide by 9, giving 20/9-7/3-29/9. Thus it reduces to the above equations with n=2/3, but cannot be generated by your cases with n an integer.

I'm surprised that no one else here knows (or knew until snwflake went hunting) of the very simple formula for generating all of the triplets: a=u^2-v^2, b=2uv, c=u^2+v^2 for integers u>v>0. That's how I came up with my counterexample 20-21-29; I just tested values of u and v until fidning one (5 and 2) that worked.

[Snowflake]

I've got them all now, I believe.

[SpencerH]

Originally posted by Snowflake
Confidence intervals will not give you much more info in this case, I believe. He has good statistical inferences (low t-score, high r square, etc.), the problem is that with only a few observations in the sample, how much could we trust those statistical inferences.

In theory t-test is applicable to small numbers but the problem is that its difficult to decide whether the data violate the t-test assumptions. IIRC violating the assumptions reduces the sensitivity of the test but doesnt necessarily invalidate results where the null hypothesis is rejected.

I still havent found a good answer so I guess I'll throw that point back to the journal to decide.

[Snowflake]

Give me a natural number, and I'll give you all the possible pythagorean triplets where this number is the shortest side.

[Snowflake]

Originally posted by SpencerH


In theory t-test is applicable to small numbers but the problem is that its difficult to decide whether the data violate the t-test assumptions. IIRC violating the assumptions reduces the sensitivity of the test but doesnt necessarily invalidate results where the null hypothesis is rejected.

I still havent found a good answer so I guess I'll throw that point back to the journal to decide.

This is why I love math better than statistics. You have so many assumtions to remember before you can do anything. Unlike math, the basis is simple, and you can derive all kind of things from it.