Of course, my favorite prime numbers website is The Prime Number Shitting Bear. He just goes on and on. It’s great!
<sigh> I need to get a life.
OK, I just wasn’t 100% sure that’s what you meant.
3 is irreducible but, according to your convoluted & unnecessary formula, not prime?
Ok, I see what you’re saying. But this distinction of Gaussian primes is an artifact of the system you use to express complex numbers. 2+(-5)[sup]1/2[/sup] has an absolute value of 3, & could be expressed as 3 at a given angle. Thus, it really is a function of 3, not something fully independent of it.
Ok, you got me, I’m totally confused by this Prime Number thing. After spending a while Googling, every place I go says that a Prime number is a number with exactly 2 factors, and that’s it. By this definition, 3 would be prime.
Now, can somebody explain why 3 is or isn’t a prime, and what the heck is this?
Granted, it’s been a while since my advanced mathematics courses in college, but this should not be this confusing 
BTW, I’m not even going to attempt to figure out on my own what the squareroot of -5 has to do with any of this!
3 is a prime integer. It is also a Gaussian prime (3 + 0i).
Frankly, I think Tenebras made another boo-boo there, but I wasn’t going to harp on it.
Well, 3 is a prime. Tenebras’s point (I believe) is that if one goes beyond integers, then 3 = 1.5 * 2, for example.
My view of the excellent OP rant is that mathematics is, in some ways, excruciatingly boring. Young nerds are taught that there’s something wonderful and fulfilling about mathematical structures like primes or groups or standard deviations. Of course, standard deviations aren’t nearly as interesting as real deviations.
Well, I guess I’ll just be hauling my non-math-knowing ass right on out of here…
Ok, so now that I seem to have confused everybody, let me make clear:
First, what is a prime? A prime is a number which is not a unit (so it doesn’t have a multiplicative inverse) and has the devisibility property that Superman desires so much. If p divides a*b, then p divides a or p divides b.
In the integers, this is equivalent to being irreducible (ie, only being divisible by 1 and iteself). I went googling myself, and was dismayed at how few websites give the real definition of prime. It makes me sad, because there are number systems where being prime and being irreducible are not equivalent. For example, Z adjoined the square root of -5.
Let’s call that Z[sub]-5[/sub]. It is everything of the form:
a + b*(-5)[sup]1/2[/sup] = a + b*(5)[sup]1/2[/sup]*i
Where a and b are integers. This is related to the Gaussian integers (complex numbers a + bi where a and b are integers) but they are not the same.
In Z[sub]-5[/sub], 3 is irreducible. I’m not going to check this here, since it’s tricky to type, but easy to do. I’ll describe how if anybody really wants to see. (We use the norm. Yay!) But 3 is not prime, because 3 divides 9, but 9 = (2 + (-5)[sup]1/2[/sup])*(2 - (-5)[sup]1/2[/sup]), and 3 does not divide either of those numbers.
Ok, I think that’s reasonably clear (like mud, most likely). Now the question is, why did I bring any of this up at all? Well, I had to go to a ring where being irreducible and being prime aren’t the same thing, and I like Z[sub]-5[/sub] because it’s relatively easy to work with and being prime and being irreducible are not the same. (On account of being not a UFD, for those Algebra fans out there)
And while 3 is irreducible in the Gaussian integers, notice that 5 is not (since (1+2i)(1-2i) = 5). But we’re not in the Gaussian integers, we’re in Z[sub]-5[/sub]. (Take that, Spiritus! 
)
So, foolsguinea, 3 is irreducible and 3 is not prime, in Z[sub]-5[/sub].
cheesesteak, the division thing isn’t unnecessary. It’s just a bit clunky, because you’re used to being in the integers, where irreducible and prime are the same thing. Take an analagy: renates are animals with kidneys, cordates are animals with hearts. It happens that all renates have hearts and all cordates have kidneys. But renate does not mean “an animal with a heart”. Likewise, in the integers, prime implies the division check, and irreducible means has no proper divisors. They happen to be the same, in the integers. But if you move to some other system, they are not the same (Z[sub]-5[/sub], for example). Just like if we were someday visited by aliens who had hearts but no kidneys, we wouldn’t call them renates, we would call them cordates. What has happened is that since everything works so well in the integers, the two have become interchangable.
december: Sort of, but not quite. When I wrote the OP, I forgot that primes must be non-units. So 15, in the reals, satisfies the division property. But it’s a unit, since 15*(1/15) = 1. And it’s not that mathematics is boring. There’s a reason why you use the division property instead of irreducibility to classify primes. There are a lot of really neat properties that primes have, no matter what ring you are in, that irreducibles don’t have. What I have a problem with is that these concepts get simplified by people who ought to know better. Heck, even I slipped on the unit thing.
Ok, I think that’s all.
Tenebras
Whenever I set a combination lock number, I generally choose prime numbers. Easy to remember, and it drives my wife nuts.
I just have to remember - did I set this to the least 3 digit prime, or did I choose the first three prime digits?
But, so far, no luck as a lottery strategy.
I like primes because of their applications in cryptography and other forms of non-obvious communication. To wit:
Hello = 2^8 * 3^5 * 5^12 * 7^12 * 11^15 = A bignum that would be a royal pain to factor but would yield the information to anyone sufficiently determined to do it.
Having a unique integer representation of all possible information, via the magic of Goedel-numbering, intrigues me to no end.
Kind of simple in a mind-crushing way, like a big rock.
The fact that there is no quick way to factor large numbers is the foundation of the strength of modern cryptography. It has placed the modern science outside of the realm of the scruffy genius, the Turing or the Babbage, and into the realm of the supercomputer and the brute-force algorithm. No longer can one do, either create or break, encryption with pen, paper, and a crossword-loving mind. That makes a part of me sad.
Of course, quantum computing will change cryptography again: Factoring huge numbers will be trivial, but encryption it is physically impossible to break, or even undetectably intercept, will be the new norm. The ball will remain in the codemaker’s court, where it has been since asymmetric-key encryption.
Would broadcasting the sequence of primes to the universe only work if the recipient have 10 digits?
In other words, do our sequence of primes only work in base 10?
Wouldn’t base 8 have a different sequence of prime numbers?
No, different bases are simply different notations used to represent the same numbers–the mathematics is unchanged. For more info, you might want to check this recent thread in GQ:
http://boards.straightdope.com/sdmb/showthread.php?threadid=104914
Ok, I realize that I’m resurrecting a dead thread. Sorry – I got the link from a different pit thread.
With that apology out of the way… I’ve been studying cryptology, but I don’t have a strong background in math. So while I appreciate the importance of prime numbers, one of the equations in this thread seemed to contradict my understanding of prime numbers (to me, 9 is not a prime number because 3 x 3 = 9, but 3 is prime because only 3 x 1 = 3). I’ve read all the posts in here, but still can’t seem to resolve this:
I understand that
But can someone explain how this disproves that 3 is a prime? I thought that you had to factor whole (integer) numbers to decide whether something is a prime. I read Tenebras’ last post in this thread, and I still don’t see how anything to do with Z(-5) is able to disprove that 3 is a prime number. Am I being obtuse? Please go step-by-step. Thank you.
Starbury, read (or reread) the whole thread carefully, the entire thread is about confusion over what a prime number actually is.
Basically, the actual definition is, in an integral domain R, p is prime if it’s not zero and not a unit, and p divides ab implies p divides a or p divides b.
The important thing being the context of whatever integral domain you’re in.
In the integral domain of integers, this definition is equivalent to what every body is used to–a prime is a number that’s divisible only by one and itself (and their negatives). Three is a prime here for the reason you mention.
In the integral domain of the real numbers, there are no primes, since everything is either a zero or a unit (every (non-zero) thing divides everything else). Three is not a prime here because it’s a unit: 3*(1/3) = 1.
In the integral domain of Z[(-5)[sup]1/2[/sup]] (which consists of numbers of the form a + b(-5)[sup]1/2[/sup], a and b integers), three is not a prime, since while 3 divides
(2 + (-5)[sup]1/2[/sup])(2 - (-5)[sup]1/2[/sup]) (which is 9),
3 does not divide (2 + (-5)[sup]1/2[/sup]) and 3 does not divide (2 - (-5)[sup]1/2[/sup]).
It’s all about context–what domain of numbers you are working in.
You written you all down on my loser list. Well, to be fair, a few of you were already on it, so I just put a star by your names.
no no no! I did NOT mean to hit submit, I was still editing!
Fuck