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extropalopakettle
No offense, but....

 Posted: Sat Jan 19, 2013 5:20 pm    Post subject: 1 Sort of a flip side to What is your favorite elegant/beautiful mathematical proof?, what is your favorite, perhaps seemingly simple, unsolved mathematical problem. It's more of a flip side to a "favorite elegant/beautiful mathematical proof" when it's a mathematical proposition believed true but which eludes proof (like, until recently, Fermat's Last Theorem). I'll start with this one: While it has been proved there are an infinite number of perfect numbers, it hasn't been proven they're all even. An integer N is a perfect number if it is equal to the sum of its divisors (other than N itself), such as 6=1+2+3, 28=1+2+4+7+14. A lot has been proved about perfect numbers, just not the seemingly simple "they're all even".
Zag
Unintentionally offensive old coot

 Posted: Sat Jan 19, 2013 5:38 pm    Post subject: 2 Ok. Looking into this one put a new one for me on the other list. I'll go post it now.
Chuck
Daedalian Member

 Posted: Sat Jan 19, 2013 7:03 pm    Post subject: 3 I didn't think there was yet proof that there are an infinite number of even perfect numbers.
extropalopakettle
No offense, but....

 Posted: Sat Jan 19, 2013 7:18 pm    Post subject: 4 Actually, it got me looking at "sum of divisors" just now. Let s(n) = sum of divisors d of n, d < n. Look at n, s(n), s(s(n)) .... etc. Starting with n=220 or n=276 is interesting ...Last edited by extropalopakettle on Sat Jan 19, 2013 7:31 pm; edited 1 time in total
extropalopakettle
No offense, but....

Posted: Sat Jan 19, 2013 7:22 pm    Post subject: 5

 Chuck wrote: I didn't think there was yet proof that there are an infinite number of even perfect numbers.

Ha ... you're right (also no proof there are an infinite number of Mersenne primes). Well, we know of more even ones than odd ones, anyway.
extropalopakettle
No offense, but....

Posted: Sat Jan 19, 2013 7:33 pm    Post subject: 6

 extropalopakettle wrote: Actually, it got me looking at "sum of divisors" just now. Let s(n) = sum of divisors d of n, d < n. Look at n, s(n), s(s(n)) .... etc. Starting with n=220 or n=276 is interesting ...

It has a name: http://en.wikipedia.org/wiki/Aliquot_sequence
Thok
Oh, foe, the cursed teeth!

Posted: Sat Jan 19, 2013 8:02 pm    Post subject: 7

 extropalopakettle wrote: It has a name: http://en.wikipedia.org/wiki/Aliquot_sequence

Incidentally, this is the mathematics behind this xkcd comic.

As for fun unsolved problems, I'll list Goldbach's conjecture: every even number >2 is a sum of two primes. The best results so far is
1. Every even number is a sum of a prime and a number with at most two distinct prime factors.
2. Every even number is a sum of at most 6 primes.
Amb
Amb the Hitched.

 Posted: Sat Jan 19, 2013 8:51 pm    Post subject: 8 Interestingly - the list here: http://en.wikipedia.org/wiki/List_of_perfect_numbers doesn't show just even numbers, all them end in 6 or 8. Maybe the proof should actually show that 6 and 8 are the only numbers perfect numbers can end in.
Jedo the Jedi
Paragon in Training

Posted: Sat Jan 19, 2013 9:38 pm    Post subject: 9

 Thok wrote: As for fun unsolved problems, I'll list Goldbach's conjecture: every even number >2 is a sum of two primes.

This was the one I referenced in the other thread, but I had no idea what it was called. mith told me about it when he was my teacher, and I've always wondered if anything had come about with it. I guess not.
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extropalopakettle
No offense, but....

Posted: Sat Jan 19, 2013 9:46 pm    Post subject: 10

 Amb wrote: Interestingly - the list here: http://en.wikipedia.org/wiki/List_of_perfect_numbers doesn't show just even numbers, all them end in 6 or 8. Maybe the proof should actually show that 6 and 8 are the only numbers perfect numbers can end in.

They've proved that all even perfect numbers end in 6 or 8.
extropalopakettle
No offense, but....

Posted: Sat Jan 19, 2013 10:04 pm    Post subject: 11

 Thok wrote: ... Goldbach's conjecture: every even number >2 is a sum of two primes.

Related to that is "Goldbach's Comet", the plot of the function g(n) for even n. g(n) = the number of ways n can be represented as the sum of two primes. Goldbach's conjectures is simply g(n)>0 (for even n).

It's hard to imagine there's no trend here:

The Potter
Feat of Clay

 Posted: Sat Jan 19, 2013 11:47 pm    Post subject: 12 Another set that has a similar plot is the number of times it takes for a number to reach 1 when: Even, ÷2 Odd, *3+1_________________Artwork | Fractals | Don't ignore your dreams; don't work too much; say what you think; cultivate friendships; be happy.
Thok
Oh, foe, the cursed teeth!

Posted: Sun Jan 20, 2013 12:21 am    Post subject: 13

 The Potter wrote: Another set that has a similar plot is the number of times it takes for a number to reach 1 when: Even, ÷2 Odd, *3+1

This is called the Collatz conjecture. It's still open.
Nsof
Daedalian Member

 Posted: Sun Jan 20, 2013 12:42 am    Post subject: 14 An obvious one: P=NP_________________Will sell this place for beer
extro...*
Guest

 Posted: Wed Feb 06, 2013 7:16 pm    Post subject: 15 Another record Mersenne prime (prime of form 2 n -1) was found yesterday: http://primes.utm.edu/primes/page.php?id=111120
extro...*
Guest

 Posted: Thu Feb 07, 2013 2:07 pm    Post subject: 16 I think it's unproved whether the Mandelbrot set (it's rational subset) is computable. (Its complement is recursively enumerable, so M is computable if it's recursively enumerable)
Death Mage
Raving Lunatic

Posted: Thu Feb 07, 2013 11:43 pm    Post subject: 17

 Thok wrote: As for fun unsolved problems, I'll list Goldbach's conjecture: every even number >2 is a sum of two primes.

Um... 3. 3 is not the sum of two primes. 1 is not a prime number.
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LordKinbote
Daedalian Member

Posted: Thu Feb 07, 2013 11:47 pm    Post subject: 18

Death Mage wrote:
 Thok wrote: As for fun unsolved problems, I'll list Goldbach's conjecture: every even number >2 is a sum of two primes.

Um... 3. 3 is not the sum of two primes. 1 is not a prime number.

Um... even.
Death Mage
Raving Lunatic

 Posted: Thu Feb 07, 2013 11:56 pm    Post subject: 19 Yea, missed that word. *shrug* Simple checking quickly made it clear that odd numbers easilly fall short of the rule. You could still claim 4, since it's the sum of the same prime twice, but meh. However... is -1 prime?_________________* These senseless ramblings brought to you by Insanity™. If you just can't figure the dang thing out, it must be Insanity™. [YOUR AD HERE!]
LordKinbote
Daedalian Member

Posted: Fri Feb 08, 2013 12:07 am    Post subject: 20

 Death Mage wrote: Yea, missed that word. *shrug* Simple checking quickly made it clear that odd numbers easilly fall short of the rule. You could still claim 4, since it's the sum of the same prime twice, but meh. However... is -1 prime?

Well, no, you couldn't claim 4 because Goldbach's conjecture makes no mention of unique primes. It's very specifically worded, and is going to hold up to more than gentle prodding for a counterexample.

Sorry, DM, but you're not going to come in here and point out the simple thing that thousands of other mathematicians seemed to miss. "Oh! This guy on the internet noticed 4 is the sum of the *same* prime! Guys, why did no one check 4?"
Thok
Oh, foe, the cursed teeth!

 Posted: Sat Feb 09, 2013 4:02 am    Post subject: 21 I find it useful to mostly ignore everything Death Mage says. That said, here prime numbers are restricted to be positive integers. In any case, -1 wouldn't be considered a prime or a composite but rather a unit, just like 1.
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