# THE OLDEST PROBLEM ON BRILLIANT...

...that is yet unsolved.

I joined almost at launch, and things were RADICALLY different back then in terms of the website's interface. One of the differences is a forums section (which was a pretty awesome feature, I don't know why anyone decided to take it down), and one of the threads contained a cool problem:

Diverges or Converges?

$\displaystyle \sum _{ n=0 }^{ \infty }{ \frac {(-1)^n \tau(2n+1) }{2n+1 } } ,$

where $\tau(N)$ denotes the number of positive integer divisors of N.

I have the answer. I also have two solutions. However, what is missing is a proof of the answer.

This seems to be a hard-core Number Theory problem. So whoever likes these, you're more than welcome to try this problem!

I shall post the answers I have if necessary. But first, give it your own shot.

Good luck!

5 years, 6 months ago

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It sure seems to converge to 1.5708...., which looks a lot like $\dfrac { \pi }{ 2 }$ ...., but I have no proof. It'd be fascinating if it really is that.

- 5 years, 3 months ago

Honestly, I did not expect this problem to blow up after two months. But just to line things up, no it does not converge to that. Good try though.

- 5 years, 3 months ago

Okay, better define exactly what "number of positive integer divisors of N" means, because I've numerically worked this out for n up to 200,000, and that's the number I'm getting and it looks pretty stable, i.e., very little difference between n = 100,000 and n = 200,000. Also, I'm adding the series in pairs, i.e., eliminating this alternating jigger.

For example, 13 would have 2 divisors, while 12 would have 6. Yes or no?

- 5 years, 3 months ago

Ugh I don't know, actually. But I can tell you this: if you subtract .9 from your answer, you'd be pretty close to the answer I have.

Oh, and for the record: the solutions look MAD complicated! Like, alien complicated.

- 5 years, 3 months ago

Okay, once in a while, I get egg all over my face, and this is one of those times. No, it's not anything like $\dfrac { \pi }{ 2 }$, in fact, I'm not sure now if it even converges. I'll have another look at this.

I was too distracted with another problem I was working on. So, finally, I've decided to tackle a few electricity problems, and I think I'm doing all right with those. Even though I still don't like it much.

- 5 years, 3 months ago

lol LOL LOL!?!? WHAT??? LOL!!!

alrighty.

LOL!!!

s

- 5 years, 3 months ago

You know, you really come up with the coolest images and GIFs

- 5 years, 3 months ago

And yet still no upvote from the humble Mr. Mendrin!

Dam I gotta write like a Nobel-Peace Prize quality comment to get an upvote from you.

And that makes it only so much precious ;p

- 5 years, 3 months ago

Okay, okay, you've just been upvoted. This problem is kind of interesting, even though number theory isn't my thing either. I'll go sleep on it.

- 5 years, 3 months ago

Congratz on

$55555555555555555555555555555555555555555555555555555555555555555555555555555!$

(I'll bet $5\$ that you read that as a factorial ;))

- 5 years, 3 months ago

Dang.

- 5 years, 3 months ago

Im getting something close to $\frac{1-\sqrt(5)}{2}$

- 4 years, 11 months ago

The Golden Ratio? Wow! You're almost there! 0.00002% error! A hint: the expression involves $\pi$.

- 4 years, 11 months ago

John, by all means, if the solution involves $\pi$, please post it for sure tomorrow, which is 3/14/15! Don't miss this wonderful opportunity to do so on such a significant day. Then I'll reshare it tomorrow.

- 4 years, 11 months ago

2morrow I get declined to MIT <.>

- 4 years, 11 months ago

Tomorrow I'm going to lose the lottery? What are you saying?

- 4 years, 11 months ago

yup...... their loss.

Anyway, I've been looking for this notebook I had solutions in for about an hour, and it's gone! Dunno what happened to it, but there I had lots of other cool stuff and I hope to find it someday.

When I do, I'll post it up right away.

Cheers

- 4 years, 11 months ago

- 5 years, 3 months ago

It probably does but it's really old and I forgot what was the name of the thread so... I just posted it all over.

- 5 years, 3 months ago

Do you have a link to the solutions?

- 5 years, 3 months ago

Nah - hence my 2nd to last line in the note.

- 5 years, 3 months ago

Please post the solutions, it's pi day after all!

- 4 years, 11 months ago

Isn't it 61685?

- 1 year, 2 months ago