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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!

Note by John Muradeli
3 years, 1 month 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. Michael Mendrin · 2 years, 11 months ago

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@Michael Mendrin Im getting something close to \(\frac{1-\sqrt(5)}{2}\) Julian Poon · 2 years, 6 months ago

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@Julian Poon The Golden Ratio? Wow! You're almost there! 0.00002% error! A hint: the expression involves \(\pi\). John Muradeli · 2 years, 6 months ago

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@John Muradeli 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. Michael Mendrin · 2 years, 6 months ago

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@Michael Mendrin 2morrow I get declined to MIT <.> John Muradeli · 2 years, 6 months ago

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@John Muradeli Tomorrow I'm going to lose the lottery? What are you saying? Michael Mendrin · 2 years, 6 months ago

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@Michael Mendrin 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 John Muradeli · 2 years, 6 months ago

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@Michael Mendrin 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. John Muradeli · 2 years, 11 months ago

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@John Muradeli 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? Michael Mendrin · 2 years, 11 months ago

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@Michael Mendrin 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. John Muradeli · 2 years, 11 months ago

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@John Muradeli 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. Michael Mendrin · 2 years, 11 months ago

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@Michael Mendrin lol LOL LOL!?!? WHAT??? LOL!!!

alrighty.

LOL!!!

s

s

John Muradeli · 2 years, 11 months ago

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@John Muradeli You know, you really come up with the coolest images and GIFs Michael Mendrin · 2 years, 11 months ago

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@Michael Mendrin 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 John Muradeli · 2 years, 11 months ago

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@John Muradeli 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. Michael Mendrin · 2 years, 11 months ago

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@Michael Mendrin Congratz on

\[55555555555555555555555555555555555555555555555555555555555555555555555555555!\]

(I'll bet \(5$\) that you read that as a factorial ;)) John Muradeli · 2 years, 11 months ago

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@John Muradeli Dang. Julian Poon · 2 years, 11 months ago

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Wait...that thread doesn't exist anymore? Bogdan Simeonov · 2 years, 11 months ago

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@Bogdan Simeonov 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. John Muradeli · 2 years, 11 months ago

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@John Muradeli Do you have a link to the solutions? Bogdan Simeonov · 2 years, 11 months ago

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@Bogdan Simeonov Nah - hence my 2nd to last line in the note. John Muradeli · 2 years, 11 months ago

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@John Muradeli Please post the solutions, it's pi day after all! Bogdan Simeonov · 2 years, 6 months ago

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