Ok, I know that I’m going to come across as a hopeless nerd here, but I’m sick and tired of people talking about how fabulous and wonderful the prime numbers are. I mean, I admit that they are cool, what with their ability to divide at least one of the factors of any number that they themselves divide. Superman weeps, he’s so jealous. “If only I could divide at least one of the factors of any number that I divide! Instead I have been cursed with the ability to fly and see through women’s clothing! Curse these trifling powers, curse them!” he moans pitifully to himself as he cries himself to sleep.
But there is no reason to think that prime numbers are some sort of universal entities. They most definately are not. 3 is not prime! There, I said it. I feel better now. 3 is prime in the integers, but who cares? 15 is prime! I shout to the unwashed masses, because today I choose to work in the reals. I like the reals. They feel so grounded, and real. That bold R just jumps out at me and says “Pay attention! I’m talking to you!” and I need that kind of security right now.
From now on, I think that I shall work in rings that aren’t unique factorization domains. Who needs them? Unique factorization is a drag. So what if I want to think of 9 as 2 + (-5)[sup]1/2[/sup] times 2 - (-5)[sup]1/2[/sup] today and 3 times 3 tomorrow. Does this make me a bad person? Does it? I don’t think so, but then, I don’t think that 3 is prime, so who cares what I think? I hope that aliens have been trying to contact us for years, sending signals which encode Gaussian primes, those heartless, green blooded ETs, don’t even get me started on them. Figure that one out, Jodie Foster, I dare you!
Wow. What a nerd herd. A thought though… perhaps you were only kidding, and obviously you know your algebra to SOME extent (you mean a ring is different from a circle?).
You would do well to stick with unique factorizaion domains, but you’re going to eat shit when someone tries to get at your credit card number, because RSA ain’t going to help you, and Rabin sure as hell isn’t going to help you.
You’ve got to admit… primes have some pretty neat properties… predictable density, yet no fast way to find them. They generate isomorphism groups for finite fields. Most of them are odd. The list goes on and on.
I remember an episode of Star Trek where one of the characters taps out the first 5 or so prime numbers to communicate something. I wonder if intelligent aliens would recognize this sequqence.
As I recall, 15 is not prime in the reals, as it has an inverse, and no unit is prime. Other than that, a beautiful rant, although even I am somewhat mystified by the choice of subject material.
As long as they’re alien scientists, and not alien lawyers.
It’s hard to envision a mathematics developing in any society, no matter how different, without starting off with counting. Counting produces the positive integers, and from there, one step at a time, you get everything else that we play around with, number-wise.
Unless they’ve got a hell of a lot more brainpower than we do, the reals are just too big to really play with. Most of the reals we actually consider are countable subsets of the reals (like the roots of polynomials with integer coefficients). And if you’re going to stay within the bounds of countability anyway, chances are you’re going to spend a lot of time with the integers. So their properties will be somewhat familiar.
But great Pit rant. UFDs - who’d’a thunk it?! Only here. I love this place.
I heard somewhere that a prime sequence is one of the few sequences that doesn’t occur naturally. That is, scientists have found some phenomena in nature that radiated gamma rays, or something analoguous in arithmatic, geometric, exponential, even Fibonacci sequences. But, we haven’t yet found anything in nature that reproduces the 2,3,5,7,11,13,17 sequence yet.
If we assume that aliens have had the same amount of luck in finding primes in nature, the easiest way to communicate that we are an intelligent species is to send a sequence of prime numbers. Thus, the Star Trek dealie makes sense!
Was it Carl Sagan who said this? Stephen Hawking (the world’s smartest man!)? I don’t remember.
It seems to me that primes are special, even if only for the reasons above.
I managed to make it through half of a book (The Mathematical Tourist by Ivars Peterson) before scouting ahead to see if it was about anything but prime numbers. Nope; didn’t look like it. As the author failed to bother to make any argument whatsoever why I, or anyone, should be interested in primes at all, much less to his obsessive degree, besides the “because they’re cool!” argument mentioned here, I threw the book at the wall.
I like the primes myself. My favorite number is 17 and I also like 23 and 47. But I’ll give you 12, whose high number of factors make base-12 the most logical base to do arithmetic in.
It was Carl Sagan. He mentioned it in Cosmos, and it became the contact method used by the Vegans (people from the Vega solar system, not vegetarians ;)) in Contact.
Ok, color me confused but I don’t understand what you mean by 15 being prime in the reals. I mean 15 is a real, 3 is a real, and 5 is a real.(As I remember it reals were numbers that when you squared them always resulted in a non negative value.) So by what I’ve just written 15 isn’t prime in reals because there’s other reals(besides 1 and itself) that can divide it.
Hmmm – don’t have nay number theiory texts handy, and I really don’t recall any definition of primes in the reals which did not point to the prime integers. Are you by any chance talking about Gaussian primes?
a + bi is a Gaussian prime if it cannot be factored into Gaussian integers other than itself and 1. A Gaussian integer is a complex number with integer coefficients.
Ok, so the answer to all this stuff about 15 is that it was late and I was stupid. All of this in a Contact fueled rage…
So the definition of prime is not, strictly speaking, that it is only divisible by 1 and itself. This is true in the integers (well, mostly. prime numbers are also divisible by -1, but that’s a bit much for the moment.) A number p is prime if it is not a unit (that’s the part I forgot) and if p divides a number, then p divides at least one of that number’s factors. This is not always the same as being irreducible (only having as factors 1 and itself).
For instance, in my example in the original post, 9 = 3*3 = (2 + (-5)[sup]1/2[/sup]) * (2 - (-5)[sup]1/2[/sup]) and 3 does not divide either of the factors given after the second equals sign. So 3, though irreducible, is not prime.
But, yes, I screwed up with the thing about 15. Mea culpa. I thought we were getting a bit too close to finishing up on the ignorance thing and decided to throw a wrench in the works. Because I’m evil.
Tenebras
Also, about the Mathematical Tourist thing, isn’t the whole book about why things like prime numbers are cool?
1 being the mos common example. I do still wonder, though, were you talking about Gaussian Primes or did you mean something else by “prime in the reals”?
Actually, every prime is irreducible, but not every irreducible is prime. – ultrafilter
Yeah, that’s what I’m saying. Irreducible and prime are not the same thing. I think I should have reordered the sentences, but anyway, yes. We agree. And Spiritus, there is no explanation. I was being a dumbass.
