Triangular Numbers

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BlueNight
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Triangular Numbers

Postby BlueNight » Sun Jan 25, 2009 10:01 am UTC

I have become enamored with triangular numbers, due to a certain property they do not share with primes or powers of 2, and more recently due to their usefulness in certain areas of my job (stacking rolls of paper).

What about triangular numbers stands out to you?
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BlueNight

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Re: Triangular Numbers

Postby The Mad Scientist » Sun Jan 25, 2009 11:18 am UTC

I just read the Wikipedia article on them and the fact that the sum of two consecutive triangular numbers is a square is nice, though obvious.

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t0rajir0u
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Re: Triangular Numbers

Postby t0rajir0u » Sun Jan 25, 2009 4:36 pm UTC

BlueNight wrote:a certain property they do not share with primes or powers of 2

Details?

The triangular numbers are interesting because they generalize in two different ways: the polygonal numbers and the other diagonals in Pascal's triangle. There's something to be said for the former, but the latter is much more interesting: it leads to a generating function you don't see very often and the solution to the indistinguishable balls, distinguishable urns problem.

The former isn't that interesting because Fermat's polygonal number theorem has a much stronger (pair of) generalizations: the 15 and 290 theorems.

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Re: Triangular Numbers

Postby qinwamascot » Sun Jan 25, 2009 8:04 pm UTC

I find triangular squares (numbers that are both triangular and perfect squares) to be interesting. They share connections with several number theory ideas, some of which are not immediately obvious. For example, if we take the ratio of successive triangular squares, it converges to 17+12sqrt(2). The sequence of triangular squares is OEIS A001110.
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Sungura
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Re: Triangular Numbers

Postby Sungura » Mon Jan 26, 2009 6:00 pm UTC

I like them because it's something I can picture. If I can picture something in math I understand it a lot better.
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BlueNight
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Re: Triangular Numbers

Postby BlueNight » Tue Jan 27, 2009 7:01 am UTC

t0rajir0u wrote:
BlueNight wrote:a certain property they do not share with primes or powers of 2

Details?

I really would rather write a paper before disclosing the details, but suffice it to say I discovered it in fifth grade, but didn't realize until the tenth grade that I had a new method of finding prime numbers. I have found exactly two mentions of this property on Teh Web, and neither is as well developed.
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antonfire
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Re: Triangular Numbers

Postby antonfire » Tue Jan 27, 2009 8:54 am UTC

Either you mentioned it in your original post because you wanted people to ask so you could tell them about it, or you mentioned it in your original post because you wanted people to ask so you could refuse to tell them about it.

Both kind of suck, but the second one sucks more, so, out with it!
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Re: Triangular Numbers

Postby PM 2Ring » Tue Jan 27, 2009 4:08 pm UTC

qinwamascot wrote: if we take the ratio of successive triangular squares, it converges to 17+12sqrt(2).


How curious! Anyone familiar with the continued fraction of sqrt(2) will recognize 17/12 as a handy approximation of
sqrt(2). Or for those who are more geometrically inclined, a 12, 12, 17 triangle is an isosceles triangle which is almost a right-angled triangle.

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t0rajir0u
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Re: Triangular Numbers

Postby t0rajir0u » Tue Jan 27, 2009 6:07 pm UTC

BlueNight wrote: I discovered it in fifth grade

Based on this comment and your completely off-base comment in the other thread about twin primes, I'm going to go out on a limb here and say that your idea doesn't actually work as well as current methods to find probable primes, which you're probably not familiar with, so either you can tell us and we can tell you if your idea works or not or you can keep it to yourself and I'll assume it doesn't.

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qinwamascot
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Re: Triangular Numbers

Postby qinwamascot » Wed Jan 28, 2009 9:08 am UTC

PM 2Ring wrote:How curious! Anyone familiar with the continued fraction of sqrt(2) will recognize 17/12 as a handy approximation of
sqrt(2). Or for those who are more geometrically inclined, a 12, 12, 17 triangle is an isosceles triangle which is almost a right-angled triangle.


Actually, continued fractions come up more than once in square triangular numbers. For example, if we have s^2=(t^2+t)/2 then the ratio t/s as we increase s are the rational convergents for sqrt(2). It's not that surprising if you have a strong background in number theory, but still is impressive.
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