# Prime Numbers (Math rant. You have been warned) **URL:** \ **Category:** The BBQ Pit\ **Created:** [March 17, 2002, 8:03am UTC](https://boards.straightdope.com/t/prime-numbers-math-rant-you-have-been-warned/98718 "2002-03-17T08:03:21Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Tenebras](https://avatars.discourse-cdn.com/v4/letter/t/977dab/32.png) [@Tenebras](https://boards.straightdope.com/u/Tenebras)\ **Post date:** [March 17, 2002, 8:03am UTC](https://boards.straightdope.com/t/prime-numbers-math-rant-you-have-been-warned/98718/1 "2002-03-17T08:03:21Z")
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! Tenebras The link that started it all: [http://www.coseti.org/klaescnt.htm#The%20Prime%20Numbers](http://www.coseti.org/klaescnt.htm#The%20Prime%20Numbers) ---
**Author:** ![Erika](https://avatars.discourse-cdn.com/v4/letter/e/47e85d/32.png) [@Erika](https://boards.straightdope.com/u/Erika)\ **Post date:** [March 17, 2002, 8:26am UTC](https://boards.straightdope.com/t/prime-numbers-math-rant-you-have-been-warned/98718/2 "2002-03-17T08:26:13Z")
::applauds enthusiastically:: ---
**Author:** ![KarmaComa](https://avatars.discourse-cdn.com/v4/letter/k/e36b37/32.png) [@KarmaComa](https://boards.straightdope.com/u/KarmaComa)\ **Post date:** [March 17, 2002, 10:00am UTC](https://boards.straightdope.com/t/prime-numbers-math-rant-you-have-been-warned/98718/3 "2002-03-17T10:00:53Z")
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. ---
**Author:** ![lucwarm](https://avatars.discourse-cdn.com/v4/letter/l/e19adc/32.png) [@lucwarm](https://boards.straightdope.com/u/lucwarm)\ **Post date:** [March 17, 2002, 1:42pm UTC](https://boards.straightdope.com/t/prime-numbers-math-rant-you-have-been-warned/98718/4 "2002-03-17T13:42:13Z")
> [@](#): > > \*Originally posted by Tenebras \* > \*\* > > But there is no reason to think that prime numbers are some sort of universal entities. \*\* 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. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [March 17, 2002, 5:12pm UTC](https://boards.straightdope.com/t/prime-numbers-math-rant-you-have-been-warned/98718/5 "2002-03-17T17:12:49Z")
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. ---
**Author:** ![RTFirefly](https://avatars.discourse-cdn.com/v4/letter/r/c77e96/32.png) [@RTFirefly](https://boards.straightdope.com/u/RTFirefly)\ **Post date:** [March 17, 2002, 7:28pm UTC](https://boards.straightdope.com/t/prime-numbers-math-rant-you-have-been-warned/98718/6 "2002-03-17T19:28:50Z")
> [@](#): > > \*Originally posted by lucwarm \* > 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 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. 🙂 ---
**Author:** ![KarmaComa](https://avatars.discourse-cdn.com/v4/letter/k/e36b37/32.png) [@KarmaComa](https://boards.straightdope.com/u/KarmaComa)\ **Post date:** [March 17, 2002, 8:24pm UTC](https://boards.straightdope.com/t/prime-numbers-math-rant-you-have-been-warned/98718/7 "2002-03-17T20:24:00Z")
Maybe aliens have an uncountable number of fingers. ---
**Author:** ![Tenebras](https://avatars.discourse-cdn.com/v4/letter/t/977dab/32.png) [@Tenebras](https://boards.straightdope.com/u/Tenebras)\ **Post date:** [March 17, 2002, 10:29pm UTC](https://boards.straightdope.com/t/prime-numbers-math-rant-you-have-been-warned/98718/8 "2002-03-17T22:29:47Z")
Damn! We hates 15 even more, now. Good catch. Thanks. ---
**Author:** ![Albert\_Rose](https://avatars.discourse-cdn.com/v4/letter/a/9de053/32.png) [@Albert\_Rose](https://boards.straightdope.com/u/Albert_Rose)\ **Post date:** [March 18, 2002, 12:53am UTC](https://boards.straightdope.com/t/prime-numbers-math-rant-you-have-been-warned/98718/9 "2002-03-18T00:53:47Z")
The square root of a prime number is irrational. Prove it! ---
**Author:** ![Tenebras](https://avatars.discourse-cdn.com/v4/letter/t/977dab/32.png) [@Tenebras](https://boards.straightdope.com/u/Tenebras)\ **Post date:** [March 18, 2002, 1:29am UTC](https://boards.straightdope.com/t/prime-numbers-math-rant-you-have-been-warned/98718/10 "2002-03-18T01:29:15Z")
The square root of a prime is irrational. I assume you mean prime in the integers? I laugh at such trivial excercises. Laugh, I say, laugh. Ha! Assume to the contrary that the square root of a prime, **p** is rational. Suppose that the square root is **a** / **b** , **a** and **b** relatively prime. So **a** [sup]2[/sup] / **b** [sup]2[/sup] = **p** So **a** [sup]2[/sup] = **pb** [sup]2[/sup] So **p** divides **a** [sup]2[/sup] and hence **a** = **pc** for some **c**. But then **p** [sup]2[/sup] **c** [sup]2[/sup] = **pb** [sup]2[/sup] or **pc** [sup]2[/sup] = **b** [sup]2[/sup] And so **p** divides **b**. \ Next! Tenebras ---
**Author:** ![Soup\_du\_jour](https://avatars.discourse-cdn.com/v4/letter/s/848f3c/32.png) [@Soup\_du\_jour](https://boards.straightdope.com/u/Soup_du_jour)\ **Post date:** [March 18, 2002, 4:34am UTC](https://boards.straightdope.com/t/prime-numbers-math-rant-you-have-been-warned/98718/11 "2002-03-18T04:34:02Z")
> [@](#): > > \*Originally posted by lucwarm > > * * * > > 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.\*\* 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. -Soup ---
**Author:** ![dropzone](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dropzone/32/7515_2.png) [@dropzone](https://boards.straightdope.com/u/dropzone)\ **Post date:** [March 18, 2002, 5:05am UTC](https://boards.straightdope.com/t/prime-numbers-math-rant-you-have-been-warned/98718/12 "2002-03-18T05:05:56Z")
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. ---
**Author:** ![Palve](https://avatars.discourse-cdn.com/v4/letter/p/a698b9/32.png) [@Palve](https://boards.straightdope.com/u/Palve)\ **Post date:** [March 18, 2002, 6:42am UTC](https://boards.straightdope.com/t/prime-numbers-math-rant-you-have-been-warned/98718/13 "2002-03-18T06:42:06Z")
I like the primes myself. My favorite number is [17](http://www.vinc17.org/yp17_eng.html) and I also like [23](http://www.geocities.com/Area51/Dungeon/7293/index.html) and [47](http://www.47.net/47society/). But I’ll give you [12](http://www.dsgb.orbix.co.uk/), whose high number of factors make base-12 [the most logical base](http://www.ubergeek.org/~chris/random/base12.html) to do arithmetic in. > [@](#): > > \*Originally posted by Soup\_du\_jour \* > \*\*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 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_. ---
**Author:** ![Dave\_D](https://avatars.discourse-cdn.com/v4/letter/d/49beb7/32.png) [@Dave\_D](https://boards.straightdope.com/u/Dave_D)\ **Post date:** [March 18, 2002, 7:49am UTC](https://boards.straightdope.com/t/prime-numbers-math-rant-you-have-been-warned/98718/14 "2002-03-18T07:49:01Z")
> [@](#): > > \*Originally posted by Tenebras \* > > 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. 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. ---
**Author:** ![Spiritus\_Mundi](https://avatars.discourse-cdn.com/v4/letter/s/c68b51/32.png) [@Spiritus\_Mundi](https://boards.straightdope.com/u/Spiritus_Mundi)\ **Post date:** [March 18, 2002, 7:25pm UTC](https://boards.straightdope.com/t/prime-numbers-math-rant-you-have-been-warned/98718/15 "2002-03-18T19:25:02Z")
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](http://www.jimloy.com/algebra/gprimes.htm)? a + b_i_ 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. ---
**Author:** ![Tenebras](https://avatars.discourse-cdn.com/v4/letter/t/977dab/32.png) [@Tenebras](https://boards.straightdope.com/u/Tenebras)\ **Post date:** [March 18, 2002, 9:21pm UTC](https://boards.straightdope.com/t/prime-numbers-math-rant-you-have-been-warned/98718/16 "2002-03-18T21:21:03Z")
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? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [March 18, 2002, 10:24pm UTC](https://boards.straightdope.com/t/prime-numbers-math-rant-you-have-been-warned/98718/17 "2002-03-18T22:24:39Z")
Actually, every prime is irreducible, but not every irreducible is prime. ---
**Author:** ![Spiritus\_Mundi](https://avatars.discourse-cdn.com/v4/letter/s/c68b51/32.png) [@Spiritus\_Mundi](https://boards.straightdope.com/u/Spiritus_Mundi)\ **Post date:** [March 18, 2002, 10:39pm UTC](https://boards.straightdope.com/t/prime-numbers-math-rant-you-have-been-warned/98718/18 "2002-03-18T22:39:53Z")
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”? ---
**Author:** ![Tenebras](https://avatars.discourse-cdn.com/v4/letter/t/977dab/32.png) [@Tenebras](https://boards.straightdope.com/u/Tenebras)\ **Post date:** [March 19, 2002, 12:07am UTC](https://boards.straightdope.com/t/prime-numbers-math-rant-you-have-been-warned/98718/19 "2002-03-19T00:07:05Z")
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. 🙂 Tenebras ---
**Author:** ![Spiritus\_Mundi](https://avatars.discourse-cdn.com/v4/letter/s/c68b51/32.png) [@Spiritus\_Mundi](https://boards.straightdope.com/u/Spiritus_Mundi)\ **Post date:** [March 19, 2002, 5:25pm UTC](https://boards.straightdope.com/t/prime-numbers-math-rant-you-have-been-warned/98718/20 "2002-03-19T17:25:41Z")
Oh, no worries. Happens to all of us. Derive Fermat’s Little Theorem as penance and aall will be forgiven. 😉 [Next page](https://boards.straightdope.com/t/prime-numbers-math-rant-you-have-been-warned/98718.md?page=2)