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| May-28-05 | | Akavall: Do you guys know about the problem about pairs of prime numbers? I think it is proven that there are infintly many prime numbers. But do pairs like, 17 and 19, 29 and 31, 41 and 43, "keep exhisting" or do they stop at some point? I believe that this problems is still not solved yet. |
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| May-28-05 | | acirce: <Akavall> Yes, it is proven that there are infinitely many primes, and the proof is very simple. It is not proven that there are infinitely many twin primes, and at the moment such a proof is far away, but it is a very good guess that there are. There was an attempt to prove it but was exposed as flawed: http://arxiv.org/abs/math.NT/0405509 http://mathworld.wolfram.com/TwinPr... |
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May-28-05
 | | Sneaky: Suppose there were a largest prime number. Call it N. Now consider the much larger number N!+1, i.e. 1*2*3*4...*N+1. Now break N!+1 into its prime factors. Whether you try to divide (N!+1) by 2, or 3, or 4, or 5, ... N, you will always end up with a remainder of 1. This should be obvious, since all of these numbers divide evenly into N! therefore they all have a remainder of 1 when you divide them into N!+1. Therefore, N! + 1 does not have any number between 2 and N as a divisor. This means that either (a) There are no prime divsors at all; i.e. N! + 1 is prime, or (b) N! + 1 has a prime divisor, but it's greater than N. In either case, we know that N cannot be the largest prime number, so we have a contradiction. Thus, there is no largest prime number. QED. |
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May-28-05
| | Sneaky: My favorite infinite series is 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 ... A teacher asked the class a long time ago to figure out what this series converges to. To our astonishment we found that it doesn't converge to anything; it converges to infinity. But it approaches infinity sooooooooo slowly. One is tempted to claim that if it was any slower, it would never get there at all! If you imagine the series when you're out in the trillions, and each term is adding microscopic quantities, and getting smaller still, you'd think that real progress is not being made. And yet I tell you that if you name any number, no matter how large, this series will exceed it if you compute enough terms. I've heard it said, in some eastern religion, that there is a mountain 100 miles high, 100 miles wide, 100 miles deep, made of solid granite, isolated from the rest of the world. Almost isolated... there is in fact a tiny bird which visits the mountain every thousand years to sharpen its beak on the rocks. When this mountain is ground entirely into dust, one day of eternity has passed. Impressive as this sounds I'll put my series up against it in terms of slowness any day. |
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| May-28-05 | | acirce: <Sneaky> 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + 1/13 + 1/17 ... (the reciprocals of all prime numbers) diverges to infinity as well! |
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May-28-05
| | Sneaky: You're kidding me?! I wonder how we know that to be true. Those primes start to become as rare as chicken-teeth when you get out far enough. |
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| May-28-05 | | acirce: Apparently it follows from Mertens' Second Theorem, the proof of which I can't really relate off the top of my head. ;-) Maybe <maoam> or somebody else is willing to fill in some of the techical details if possible. http://mathworld.wolfram.com/TwinPr...
http://mathworld.wolfram.com/Merten... Much fascinating stuff. |
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| May-28-05 | | woodenbishop: The Caterpillar Lime
It will rob my innocence cold and fast
Leave me thirsty for the salt ringed glass
Sex and tequila on my mind
All I see is the caterpillar lime
Slurping, slurping
Red beans, tomatoes, and guacamole
A Tejano dish for the most holy
The cigarette taste on her lips is sublime
But the kiss is missing a caterpillar lime
Slurping, slurping
Crackled ice melts way too soon
Under immense heat of the Spanish moon
My waitress is appalled by a lewd crime
Of my insanely sucking caterpillar lime
Slurping, slurping
My desire for love the cafe devours
With every sip I dream of flowers
And the petals contain this beautiful rhyme
Fulfilling the thirst of the caterpillar lime
Slurping, slurping
-Paul Hopkins |
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| May-29-05 | | Catfriend: Yes, actually it all is a direct result of several results on series conencted to integration. |
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| May-29-05 | | Catfriend: <sneaky: Those primes start to become as rare as chicken-teeth when you get out far enough.>
The twin prime conjecture says you're wrong:) |
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| May-29-05 | | azaris: Proof of Mertens' Second Theorem:
http://front.math.ucdavis.edu/math.... |
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| May-29-05 | | Akavall: <It is not proven that there are infinitely many twin primes, and at the moment such a proof is far away, but it is a very good guess that there are.> This sounds a lot like Riemann Hypothesis, it just has to be correct, but there is still no proof. |
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| May-29-05 | | maoam: <Akavall: This sounds a lot like Riemann Hypothesis, it just has to be correct...> Me and J.E. Littlewood disagree with you! It'd be nice for the zeroes of the zeta function (or the L-series associated with any Dirichlet character) to have real part 1/2, but I don't see why it must be true. |
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| May-29-05 | | Catfriend: <maoam> Professor Haran from the Technion institurte (I had the honour to meet him personnaly) is currently developing a method probably strong enough to prove Riemann!
Besides, any evidence (not proof) we had about zeta function convinces us more and more the conjecture is a right one. Of course, it's all is just empty words till the final proof is published! |
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| May-29-05 | | maoam: <Catfriend: Besides, any evidence (not proof) we had about zeta function convinces us more and more the conjecture is a right one.> I've seen the evidence and I'm not convinced. |
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May-29-05
| | Sneaky: I saw a documentary about the guy who solved mathematics most famous unsolved problem: Fermat's Last Theorem. Yes, it's been solved, and the proof is a giant tome that weighs 5 pounds and draws upon mathematical theorems spanning across history right up to the modern day. Most of you know the story but let me recap for the few who don't--the great mathematical genius Fermat wrote in his diary that he found a wonderful proof that a^n+b^n=c^n has no integer solutions for any value of n>2, but there is not enough space in the margin, and so he would explain it the next day. Alas, Fermat died in his sleep, and for several centuries mathematicians ripped their hair out trying to find what Fermat had found. My question is this--was Fermat deluded?? Did he really have a proof which "didn't fit in the margin" but presumably would fit on a page or two of his diary? Is there an elegant proof which would allow us to dispense with this 5 pound tome, or not? Also, would a real mathematician even care...? After all, a proof's a proof! |
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| May-29-05 | | Catfriend: <maoam> Yes, of course it isn't convincing enough! That's why it's evidence, not proof! <sneaky> mostly correct, but innacurate. Wiles' proof isn't 5 tomes, "only" ~100 pages. As to Fermat and the seemingly existing proof - it's an open question, and the answer is somewhat similar to religion - you choose what to think! There is no clear evidence Fermat has any proof, and most great mathematicians failed to prove the theorem using simple tools. Who would care? That depends on waht's important to you in math! Some will say it's an art, and to them the aestethic value is the main one! Others respect the practical importance math has, and to them it's really doesn't matter! |
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| May-29-05 | | maoam: <Sneaky>
Your account of the story is correct modulo certain details (see http://www-groups.dcs.st-and.ac.uk/...). To answer your question: Fermat may not have been deluded, but he certainly did not possess a correct proof of his theorem. Regarding an elegant proof of FLT, I'll simply quote Solomon Lefschetz: "Don't come to me with your pretty proofs. We don't bother with that baby stuff around here." Once something is proven it's trivial. That didn't stop Gauss from proving the quadratic reciprocity theorem seven times; and there's no denying the beauty of say, Erdos' elementary proof of Bertrand's postulate. An elegant proof may exist, but it's probably beyond us to find it (for now at least). |
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| May-29-05 | | Catfriend: <maoam> excellent post:)
Yes, of course there are many examples of beauty with no practical meaning.
And perhaps it's the most important thing...
But that defers nothing of the respect due to Wiles! His proof is amazing, useful, revoluationary etc. ! |
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| May-29-05 | | get Reti: Speaking of proofs, heres three theorems that have never been prvoed-any even number over two can be written as the sum of two prime numbers and every composite number is equidistant from two prime number and if the three sides of a right triangle are integers, the product of the sides will be divisible by 60. |
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| May-29-05 | | Catfriend: <get Reti> The first is Goldbach's conjecture, the second is a result that can be achieved by proving it
and the last one was proved! |
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| May-29-05 | | Catfriend: It's a result of the generalised form of Pythagorean triangles. |
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| May-29-05 | | get Reti: And here's a simple proof of what snaeky said:
1+1/2+1/3+1/4+1/5+1/6+1/7+1/8.....>
1+1/2+1/2+1/4+1/4+1/4+1/4+1/8...=1+1+1.... |
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| May-29-05 | | TheSlid: I found it more elegant with just primes. |
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| May-29-05 | | Catfriend: Yes, all these converging series are analysed in the same way, it's a rather standart operation. |
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