Talk:Prime number/Draft

Article Checklist for "Prime number/Draft"

Workgroup category or categories	Mathematics Workgroup [Categories OK]
Article status	Developing article: beyond a stub, but incomplete
Underlinked article?	No
Basic cleanup done?	Yes
Checklist last edited by	Greg Woodhouse 07:08, 5 April 2007 (CDT)

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Primes and their generalizations

After some thought, I added a clarification to the introductory material. The reason is that while the rational primes (i.e., primes in $\mathbb {Z}$) are very important in cryptographic applications, other engineering applications (notably error detecting and correcting codes, where linear codes are very important) depend upon properties of primes and factorization in other rings (such as $\mathbb {F} _{2}[x]$). It may seem like a small thing, but I do want to be sure that the claims made in the introductory section are correct. Greg Woodhouse 05:41, 5 April 2007 (CDT)

Just delete this?

I noticed that someone removed the hyperlinks from the latter part of the introductory paragraph, and I agree that this was a good idea. To be honest, I wouldn't mind just deleting

Understanding properties of prime numbers and their generalizations is essential to modern cryptography, and to public key ciphers that are crucial to Internet commerce, wireless networks, telemedicine and, of course, military applications. Less well known is that other computer algorithms also depend on properties of prime numbers. These algorithms allow computers to run faster, computer networks to carry more data with a greater degree of reliability, and are basic to the operation of many consumer electronics devices, such as television sets, DVD players, GPS devices, and more. Life as we know it today would not be possible without an understanding of prime numbers.

I put it in there to provide some motivation for the study of prime numbers, but I'm not so sure I don't find it distracting (or just plain too long) without the hyperlinks. Greg Woodhouse 10:14, 5 April 2007 (CDT)

I think it's a bit too much; especially the last sentence. However, don't throw out the baby with the bath water. We do need some motivation. A simple solution would be to retain only the first sentence (personally, I'd also delete telemedicine).

I had some other comments when I read through the article. I'll just jot them down here for you to consider or ignore as you see fit. You already resolved one of them (in Euclid's proof, explain why it's impossible that no prime divides N) by adding a discussion on unique factorization.

• The aside on notation. I think the definition of prime number without symbols works perfectly fine, making me wonder why you praise the virtues of notation at that place. Incidentally, you need to explain the notation a | b.
• The equivalence of the two definitions for prime numbers is in fact quite important (unique factorization depends on it), and should perhaps be stressed more.
• What do you have in mind when you say that the second definition is preferred in advanced number theory? It's a long time ago that I looked at number theory, but I thought both were used (they are called irreducible and prime elements, respectively).
• On a first reading, the proof of unique factorization looks a bit messy, though I can't articulate exactly what the problem is. I'll try to have a look at it later.

-- Jitse Niesen 19:52, 5 April 2007 (CDT)

• I deleted the last sentence of the introduction, along with the reference to telemedicine.
• What I think I was trying to do with the notation for "divides" (not that I really planned it out in advance) is inrouce the notation by using it, and then step back and explain what it means. I'll add something there.
• The comment about the latter definition of "prime" being more characteristic of advanced work was inappropriate (and probably wrong). It's gone now. As I'm sure you realize, what I had in mind is that the concepts prime and irreducible just happen to coincide in Z because it's a PID. Right now, you're seeing my thoughts in rather raw form, and I guess I was thinking that I didn't want to get involved with a discussion of primes vs. irreducible elements, but I wanted to at least note that there is a difference.
• I don't like what I wrote about unique factorization, either. I didn't really want to dwell on it too much, but by the time I had written it out, the argument was just too long, and a bit awkward sounding. I'll see what I can do. Greg Woodhouse 23:49, 5 April 2007 (CDT)

Okay, I've rewritten the proof of prime factorization and filled in Euler's proof that there are infinitely many primes. Greg Woodhouse 08:26, 6 April 2007 (CDT)

What to include?

The topic of this article is obviously a big subject. When I picked up this article (from the "most requested" list on the WG page), I wasn't sure how much I wanted to cover, though some of the basics are clear. I at least want to state the prime number theorem, and But what about, say, say something about unsolved problems about prime numbers.But what about, say primality testing? I haven't even talked about the sieve of Eraosthenes yet! I thought about covering, say, the primes in the rings of Gaussian and Eisenstein integers, but that should probably be left to another article. What do you think? Greg Woodhouse 08:41, 6 April 2007 (CDT)

I've added a section on primality testing. I'd say: include full descriptions of both trial division and the sieve of Eratosthenes, but leave out detailed discussion of optimizations and complexity analysis (leave those aspects for subarticles). Mention that there are faster algorithms such as the Miller-Rabin test, but don't describe them in this article. I agree that generalizations of primes in other rings should largely be left to another article.

Looking at the Wikipedia article for inspiration:

• We should definitely have a section on the distribution of primes (PNT, prime counting function), on the general problem of finding patterns in the primes (mentioning Ulam's spiral, etc)
• We should describe applications of prime numbers in some more detail. This could look similar to the section in the Wikipedia article.
• We don't need trivia lists like Wikipedia's "Properties of primes" and "Primes in popular culture" and "Trivia" sections. That's not to say all of the content of those sections would be inappropriate here, but I'm sure we can come up with a more coherent article structure.
• There are lots and lots of special classes/categorizations/subsequences of primes. Except for perhaps twin primes and Mersenne primes, I think these are mostly trivia and should be left to another article.

Fredrik Johansson 10:06, 6 April 2007 (CDT)

I'll largely be echoing Fredrik here. PNT is important as a standard non-trivial result. Riemann hypothesis is worth a million bucks; need I say more? Twin prime conjecture is accessible and drives home the point that not everything in maths has been done centuries ago. I'm reserving judgement about Ulam's spiral. Primality testing, beyond Eratosthenos, should probably be kept brief: something about the quest for the largest known prime number, and the polynomial algorithm found recently by the Indians (AKS?). Generalizations should also be briefly mentioned. Perhaps a paragraph about Gaussian integers as an example. And applications; can be contrasted with Hardy's "number theory is beautiful because it's useless". All that will probably be enough. -- Jitse Niesen 12:22, 6 April 2007 (CDT)

I had never heard of Ulam's spiral, but looking at the article in Mathworld, I see Athur C. Clarke mentioned it in "The City and the Stars". It seem to be a popular culture connection (which has nothing to do with whether or not it's mathematically interesting, of course!) I'd leave it out of this article, at least for now. I'm trying to think of a good way to handle the PNT (and the Riemann hypothesis). Simply stating these results without giving any indication of their significance or how they fit in to number theory in general hardly seems enough. In my opinion, just stating results or definitions is where we move from being an encyclopedia to being a dictionary, at least so far as mathematics is concerned. Greg Woodhouse 16:28, 6 April 2007 (CDT)

Prime Number Theorem

I just added a formal statement of the theorem. Obviously, more exposition is needed.