Do you find the concept of there being an infinite number of prime numbers intuitive or unintuitive?

I have no reason for asking, other than I’m just curious to know...

10/14/2009 7:21:52 PM

http://en.wikipedia.org/wiki/Euclid%27s_theorem btw

10/14/2009 7:34:45 PM

intuitive

if there are infinite numbers, there are infinite primes.

at the same time, when i stare at a 10 million DIGIT number which is supposed to be prime, i just shake my head and think, isn't there ANY number before it that could go into it???


[Edited on October 14, 2009 at 10:28 PM. Reason : ]

10/14/2009 10:27:36 PM

intuitive

[Edited on October 14, 2009 at 10:30 PM. Reason : l]

10/14/2009 10:30:09 PM

prime numbers are like pimples on a strippers ass--you might notice them, but they don't usually matter

10/14/2009 10:35:25 PM

intuitive

if there are infinite numbers, there are infinite primes.

10/14/2009 10:39:55 PM

very intuitive if you know the p₁ * p₂ * ... * p_n + 1 proof

10/14/2009 10:40:54 PM

Quote :

" if there are infinite numbers, there are infinite primes."

that doesn't follow at all. there are many types of finite number sets in the set of all numbers

10/14/2009 10:42:27 PM

I've been a math grad student for quite awhile, and I've basically lost all intuition for things like this

my intuition has been raped too many times

10/14/2009 10:47:16 PM

^^on the surface it works, but yeah, i know what you mean.

10/14/2009 10:55:37 PM

yeah seriously, i have the mathematical intuition of hamster

well, a really smart hamster that went to college anyways

10/14/2009 10:55:43 PM

Quote :

"that doesn't follow at all. there are many types of finite number sets in the set of all numbers "

out of shear curiosity, like what?

10/14/2009 11:00:51 PM

You want something unintuitive?

Look at the cantor set:

Start with the unit interval [0,1], remove the middle third, then remove the middle thirds of the 2 remaining segments, then the middle thirds of the 4 remaining segments and so on and so forth.

at the end, you've removed a total length of 1/3 + 2/ 9 + 4/27 + 8/81 +... = 1

So you've removed a length of one, that must mean you have nothing left, right?

Wrong, you are left with all numbers that have trinary representation using only 0's and 2's (look up the proof). But! there is a 1-1 correspondence between this set and the set of all numbers that have a binary representation of 0's and 1's (map the 2's to 1's, and faneggle with the numbers that have leading zeros and infinite representations).

so what you have left is exactly as many numbers as you started with.

think about that!

10/14/2009 11:04:43 PM

^^the set of natural numbers less than 12

[Edited on October 14, 2009 at 11:05 PM. Reason : ^]

10/14/2009 11:04:51 PM

well prime numbers are just the set of numbers defined by some condition, right? that condition being it has two distinct divisors - itself and 1. just so happens this particular set end up being infinite

it's pretty easy to come up with another set of numbers defined by some other condition, say the set of natural numbers greater than 5 and less than 10. that's clearly finite. and that's all i was really saying

[Edited on October 14, 2009 at 11:05 PM. Reason : ^ exactly]

10/14/2009 11:05:22 PM

^^ the set of all natural numbers below 10

[Edited on October 14, 2009 at 11:06 PM. Reason : . dammn you fkl]

10/14/2009 11:05:57 PM

cantor set ain't nothin

even a hamster like myself knows the shit out of that

10/14/2009 11:07:34 PM

i see.

that's pretty lame. fuckin' math

10/14/2009 11:09:24 PM

alright qtmnfred, ill bring out the big guns. stop me if you've heard this one

an Isometry is a distance preserving mapping. For instance the mapping y = x + 5 is an isometry because d(x1, x2) = d(x1 + 5, x2+5) where d() is a valid distance function. In R^2 there are only a few isometries, namely, a shift (y=x+5), a flip (y=-x), and a rotation about a fixed point.

The infinite set (1,2,3,...) and the mapping (y=x+1) is interesting because it is an isometry between the set (1,2,3...) and a proper subset of the set (2,3,4...).

the set (1,2,3...) however is not bounded, (for TWW purposes, this means you can't draw a circle of finite radius around all points in the set).

However, there is a bounded set A in R^2 and an isomoetry F such that F maps A to a proper subset of A. Tell me that one qtmfredyjej

10/14/2009 11:15:52 PM

10/14/2009 11:21:05 PM

^uhh, what?

10/14/2009 11:21:40 PM

you heard me. what do i need to do, draw ya a picture?

10/14/2009 11:22:36 PM

lol, answer me!

10/14/2009 11:23:37 PM

{1}

10/14/2009 11:24:16 PM

+1

10/14/2009 11:25:03 PM

^^ What is your mapping, huh, huh?

10/14/2009 11:26:00 PM

i find the proof offered by euclid to be insubstantial

the immediate proofs called out by euler and ect to be more substantiative, but the fact remains that the one for one definitive proof does not exist. its a shame of modern mathematics and it needs to be dealt with.

10/14/2009 11:27:30 PM

shut up Fermat what do you know bout some maths?



[Edited on October 14, 2009 at 11:28 PM. Reason : ^^ take your pick son]

10/14/2009 11:28:17 PM

lol raise your hand if you don't want your wife to scan me her naughty bits

OK

there are SEVERAL primes that satisfy that original presumption

10/14/2009 11:31:57 PM

This is high school stuff at best guys.

10/14/2009 11:34:05 PM

The Banach-Tarski Theorem, the end-all and be-all counterintuitive mathematical result:

take a (solid) mathematical sphere of radius 1

it is possible, with a finite number of cuts, to cut that sphere into pieces and rearrange those pieces into two spheres of radius 1...yes, two spheres, each identical to the original

this is why I have serious doubts about the Axiom of Choice

10/14/2009 11:37:34 PM

sounds like a pretty solid bar trick

10/14/2009 11:39:14 PM

yeah I'm pretty sure this is what Jesus used for the loaves and fishes

10/14/2009 11:40:28 PM

Or, the objects with finite volume, but infinite surface area.....

It has been a long, long, long time since I have been in higher level math.

10/14/2009 11:42:54 PM

^^
you mean the fact that exaggerated rumors were a far more important form of entertainment before TV? What does that have to do with spheres?

[Edited on October 14, 2009 at 11:44 PM. Reason : -]

10/14/2009 11:43:59 PM

see, it becomes infinite truth the instant you extrude it, yet it goes against taniyama shimura conjecgture the instnant you do that. one simple assumption makes the rest unreliable in an instant and andrew wiles made damn sure there sure should be no confusion in the area of infinite primes



a legitimate question we have indeed

10/14/2009 11:47:01 PM

i once saw a proof somewhere that infinity was finite.

10/14/2009 11:58:55 PM

i once saw a proof that i'm about to add your shit to my ignore list

I'm AstralAdvent and i approved this message.

10/15/2009 12:08:01 AM

Quote :

"alright qtmnfred, ill bring out the big guns. stop me if you've heard this one an Isometry is a distance preserving mapping. For instance the mapping y = x + 5 is an isometry because d(x1, x2) = d(x1 + 5, x2+5) where d() is a valid distance function. In R^2 there are only a few isometries, namely, a shift (y=x+5), a flip (y=-x), and a rotation about a fixed point. The infinite set (1,2,3,...) and the mapping (y=x+1) is interesting because it is an isometry between the set (1,2,3...) and a proper subset of the set (2,3,4...). the set (1,2,3...) however is not bounded, (for TWW purposes, this means you can't draw a circle of finite radius around all points in the set). However, there is a bounded set A in R^2 and an isomoetry F such that F maps A to a proper subset of A. Tell me that one qtmfredyjej "

What do you mean by “R^2”?

[Edited on October 15, 2009 at 12:33 AM. Reason : i have a feeling i’m going to wake up in the morning and realize what you meant...]

10/15/2009 12:32:47 AM

if there are infinite numbers, there are infinite primes.

so with Cantor, the number actually gets smaller in a very predictable pattern that never reaches 0 even though it infinitely gets smaller...


sets get weird man, really weird http://en.wikipedia.org/wiki/Cantor_set


as for the sphere theorem: However, the pieces themselves are complicated: they are not usual solids but infinite scatterings of points.


i never like sets....

[Edited on October 15, 2009 at 12:57 AM. Reason : s]

10/15/2009 12:38:29 AM

I mean the real plane. sorry, that was poorly typed, im looking for some set of (x,y) pairs

10/15/2009 1:21:43 AM

Intuitive.

Quote :

"You want something unintuitive? Look at the cantor set: Start with the unit interval [0,1], remove the middle third, then remove the middle thirds of the 2 remaining segments, then the middle thirds of the 4 remaining segments and so on and so forth. at the end, you've removed a total length of 1/3 + 2/ 9 + 4/27 + 8/81 +... = 1 So you've removed a length of one, that must mean you have nothing left, right? Wrong, you are left with all numbers that have trinary representation using only 0's and 2's (look up the proof). But! there is a 1-1 correspondence between this set and the set of all numbers that have a binary representation of 0's and 1's (map the 2's to 1's, and faneggle with the numbers that have leading zeros and infinite representations). so what you have left is exactly as many numbers as you started with. think about that!"

It's also weird that the set of even naturals is the same size as the set of naturals. The lesson at the end of the day is that you have to be a formalist about infinity, I think.

The most counter-intuitive mathematical result I've seen, I think, is probably Goedel's incompleteness theorem. I'm not sure "counter-intuitive" is the right way to say it, but it seems like it shouldn't be true prima facie. It's just a strange, surprising result (let's put it that way).

Quote :

"The Banach-Tarski Theorem, the end-all and be-all counterintuitive mathematical result: take a (solid) mathematical sphere of radius 1 it is possible, with a finite number of cuts, to cut that sphere into pieces and rearrange those pieces into two spheres of radius 1...yes, two spheres, each identical to the original this is why I have serious doubts about the Axiom of Choice"

Fair enough. I have some doubts as well but honestly, what are you going to do? It's too damn useful to just chuck.

Then again, have you seen Shelah's proof of Morley's conjecture WITHOUT choice? I guess it's possible to recover some amount of mathematics without it, but if you have to be as smart as Shelah I'm fucked.

[Edited on October 15, 2009 at 10:26 AM. Reason : .]

10/15/2009 10:24:22 AM

http://www.networkworld.com/community/node/46184

Hows that for unintuitive

10/15/2009 5:04:51 PM

I mean it seems like if you get large enough that something should be able to divide it. But think about how gappy natural numbers are; once you fixate on that property, it makes more sense (in my opinion).

10/15/2009 5:52:09 PM

If two separate things are both infinite, can one infinite thing have more than the other?

Prime numbers are infinite. But so are non-prime numbers. Doesn't it seem like there would be more infinite non-prime numbers than infinite prime ones?

What is that answer to this?

10/15/2009 6:56:59 PM

::not a math person::

However, I think it's due to the fact that infinite being defined as "going on forever" means that you can't really have "more" or "less" or even really compare infinities because there are always more numbers. to be added to the set. As for the prime versus non-prime thing, I guess you could think of it as the primes approaching infinity at a slower rate because of the distance between each of them, but with an unbounded set like infinity even that would cease to mean anything as you approach infinity.

.02

[Edited on October 15, 2009 at 7:12 PM. Reason : oh man. sleepy]

10/15/2009 7:11:10 PM

Quote :

"Doesn't it seem like there would be more infinite non-prime numbers than infinite prime ones?"

It seems like that, but in reality there are just more non-prime numbers earlier in the sequence if that makes sense. Terms like more and less don't even apply to the concept of infinity.

[Edited on October 15, 2009 at 7:12 PM. Reason : ^yep]

10/15/2009 7:12:39 PM

^^^ Great question! Yes, there is such thing as 'more infinite', just, not in this case. the amount of primes and the amount of non primes are equal.

However, the amount of whole numbers and the amount of Real numbers are different, even though they are both infinite.

For example. there are just as many even numbers as there are natural number. Proof. divide every even number in half, and youll end up with all the natural numbers, so you must have started with all of them. to go the other way, multiply every natural number by 2 and you end up with all the even numbners.

The Real numbers are 'bigger' in a sense, that there is no mapping from the natural numbers to the real numbers, but there is a mapping from the real numbers to the naturals (because every natural is also real).

Check out cantors diagonalization argument for a proof of this

10/15/2009 7:17:10 PM

Quote :

"you can't really have "more" or "less" or even really compare infinities because there are always more numbers"

Not a true statement. There are varying degrees of infinite, each of which differ infinitely in magnitude.

http://en.wikipedia.org/wiki/Aleph_number

10/15/2009 7:20:25 PM

set em up

10/15/2009 7:21:06 PM