I'm 21 now!

[Xyzzy]

I apologize for derailing the OP's birthday thread!

I'll go read up some more on triangular numbers. My first impulse is to write an ugly Perl program to brute force something but instead I'll try to understand it mathematically.

I knew I shouldn't have trusted Wikipedia. (I picked up my original claim there. Every year I check my age to see what interesting things are associated with that number. I should have checked before posting!)

[wblipp]

Originally Posted by Mini-Geek

Because it's his age? Unless he's over 170

I checked up to 10²⁵⁰, so I already knew 36 was the only reasonable age. But this wasn't stated as a puzzle, it was stated as a mathematical fact.

Originally Posted by Mini-Geek

due to numbers being infinite, I would think that there are likely many more

I have a heuristic that suggests there are no more. But I was wondering if somebody had a proof.

[axn]

It is fairly easy to show this:

Let n(n+1)/2 = [m(m+1)/2]^2 = m^2(m+1)^2/4

i.e., n(n+1) = m^2(m+1)^2/2

But since n and (n+1) are relatively prime, we must have either

n=m^2, n+1 = (m+1)^2/2
or
n=(m+1)^2/2, n+1 = m^2
[I'll leave it as an excercise why the /2 factor must go with (m+1)^2]

The first one leads to m=1 (solution = 1, 1), the second one leads to m=3 (solution = 6, 36)

[ixfd64]

LOL, come to think of it, we're so geeky at times.

[wblipp]

Originally Posted by axn1

It is fairly easy to show this:

Let n(n+1)/2 = [m(m+1)/2]^2 = m^2(m+1)^2/4

i.e., n(n+1) = m^2(m+1)^2/2

But since n and (n+1) are relatively prime, we must have either

n=m^2, n+1 = (m+1)^2/2
or
n=(m+1)^2/2, n+1 = m^2

I don't see why you can reject the possibility that

m = pq
m+1 = st

2n = p²s²
n+1 = q²t²

OR

n = p²s²
2(n+1) = q²t²

[axn]

Originally Posted by wblipp

I don't see why you can reject the possibility that

m = pq
m+1 = st

2n = p²s²
n+1 = q²t²

OR

n = p²s²
2(n+1) = q²t²

Hmmm... Didn't think of that. Any chance of salvaging this approach, however? Some restrictions on sizes, etc?

[cheesehead]

Originally Posted by Xyzzy

I knew I shouldn't have trusted Wikipedia.

Mathworld.

http://mathworld.wolfram.com/

The self-proclaimed "Web's Most Extensive Mathematical Resource".

Remember it in your time of need.

(now that Eric Weisstein and CRC Press have settled their legal dispute about CRC Press's shameful (IMO) power play on Eric's naivete a few years ago -- see http://mathworld.wolfram.com/docs/faq.html)

- - -

From http://mathworld.wolfram.com/TriangularNumber.html:

"The numbers 1, 36, 1225, 41616, 1413721, 48024900, ... (Sloane's A001110) are square triangular numbers, i.e., numbers which are simultaneously triangular and square (Pietenpol 1962). The corresponding square roots are 1, 6, 35, 204, 1189, 6930, ... (Sloane's A001109), and the indices of the corresponding triangular numbers [T_n] are [n=1], 8, 49, 288, 1681, ... (Sloane's A001108)."

So, 1 and 36 are not the only possibilities, but Xyzzy'd have to be older than Methuselah otherwise.

[xilman]

Originally Posted by Xyzzy

How embarassing! I wonder how I should reword this? Let me go think for a while.

Ok, I've waited long enough for you to get your act together.

The smallest triangular number which is the square of a different triangular number.


Paul

[xilman]

Originally Posted by cheesehead

The corresponding square roots are 1, 6, 35, 204, 1189, 6930,
....
So, 1 and 36 are not the only possibilities, but Xyzzy'd have to be older than Methuselah otherwise.

And which of those square roots are themselves triangular?



Paul

[Xyzzy]

Originally Posted by xilman

The smallest triangular number which is the square of a different triangular number.

That'll do . That'll do.

At this point I'm just waiting for my birthday so I can put all this behind me.

[wblipp]

Originally Posted by xilman

And which of those square roots are themselves triangular?

Yes, cheesehead seems to have forgotten that the square root must also be a triangular number. As I reported in post #13, I've checked that there are no other triangular square roots up to 10²⁵⁰ (the square root this large, so the number up to 10⁵⁰⁰). I don't have a proof there are no others, though.