@FaheemMitha I think the model works in the following way: someone (or a small group) have an idea for a site. They set it up and then they have a certain amount of days to get other people to commit to the site (to show that it's viable.) Once there are enough committed users, the site goes live to public beta, and if there's enough users/interest in the site after some period of time, it becomes permanent. See area51.stackexchange.com/faq
@FaheemMitha Go to stackexchange.com/sites, and hold your mouse over where it says the age of the site. The text that pops up (and is visible in the status bar) says the dates it became a private beta, open beta, and "full" site.
TeX.SX became a private beta July 26 of last year, public beta August 2, and launched November 11.
@FaheemMitha As I understand it, the private beta is the time where people have to commit before using the site. Once the site is in public beta, anyone can use it.
"Private Beta" lasted a week and was a time when only those who had "committed" to the site had access to it. The purpose was to "seed" the site so that it already had a decent number of questions and answers for when it went "live". That way, people would be drawn to the site because they had a question that was already answered here.
"Public Beta" was something of the order of three months (don't remember exactly) and was where the site was open to all, but was going through a sort of "trial" period in the SE set up. This is where we had to prove to the SE OverLords that there was enough momentum and a big enough community that this would produce a decent resource for TeXers.
@AndreyVihrov Sorry to persist in this, but actually both "proofs by contradiction" need to assume the following result: "A number n is composite if and only if there is a prime number p strictly less than n which divides it.". In the "traditional" one, this is needed to say "If q + 1 is not prime, then there is a prime less than it which divides it, but that prime can't be one from our list.". In my version, this is needed to say that since q + 1 is not divisible by a prime less than it,
it must be prime.
So in terms of dependent results and knowledge about primes, I judge the two equivalent.
But be clear: I'm not saying that my reading is better! Yours is the clearer one as it is closer to "this" universe. (But the positive proofs are best.)
In terms of positive proofs, here's another one. We know (from that auxiliary lemma, plus a little induction) that the primes generate the natural numbers multiplicatively. To see that the natural numbers cannot be finitely generated, take a finite set, say A. Then 1 + \prod_{a \in A} a is an integer which is not in the submonoid generated by A. Hence A does not generate the natural numbers and so the natural numbers cannot be finitely generated.
I present this proof because it clearly separates the roles of everything involved. All we need to know about the primes is that they generate the natural numbers by multiplication. Once that is established, we can forget them. After that, it's all about the finiteness, and this part doesn't need primeness.
@AlanMunn: Since I'm being in prime pedantic mood, I'll just say that I'd be wary of deciding anyone's nationality by the language that they spoke. Under that rule, the aristocracy of England would have been classed as French for many, many years.
@PauloCereda: My internet.cls stuff is designed for writing new documents in LaTeX knowing that they will be put on to the web (or converted to some format) from the outset. However, it can also cope with existing documents, though they might need a little tweaking, so long as they aren't too intricate.
I came across a page that purports to be a translation of Euclid's actual proof. From this, then the positive proof is actually that supplied, not the proof by contradiction. So egreg's positive proof is the closest to what Euclid wrote.
@AndrewStacey He used the least common multiple instead of the product of the numbers in the "assigned multitude", which is "geometrically sound": for Euclid numbers were segments and such is their least common multiple.
It would have been difficult to assign a "meaning" to the product of members of any "assigned multitude". Now we don't have Euclid's problems in accepting products of any number of factors. It's nice to ask oneself why Euclid's proof doesn't work in the ring of p-adic integers, where there's only one prime.
@AndrewStacey I'll give internet.cls a try. TBH, I prefer your approach. IMHO it's wiser to try a conversion within TeX itself than relying on external tools. ;-)
\begin{document} \begin{frame} \frametitle{There Is No Largest Prime Number} \framesubtitle{Based on Euclid's Original Proof} \begin{theorem} There are more than finitely many prime numbers. \end{theorem}
\begin{proof} \begin{enumerate} \item<1-| alert@1> Let \(p_1\), \dots, \(p_n\) be prime numbers.
\item<2-> Let \(q = \lcm(p_1,\dotsc, p_n)\).
\item<3-> Let \(p\) be a prime dividing \(q + 1\).
\item<4-> If \(\exists j : p = p_j\) then \(p | q\) and \(p | q + 1\), so \(p | 1\), which is absurd.
@egreg: What do you think? I tried to keep the argument the same, but also keep the number of points down (in particular, I reduced the "If q is prime ... if q is not prime" to a single case) and make it modern in language and symbols. I deliberately kept the "which is absurd" as I think it's a fantastic phrase.
@PauloCereda Let me upload the latest version to my website ... it's undergone quite a few changes since I last posted it. What final format do you want to have?
@egreg I'd guess that "assigned multitude"s were finite. It's also interesting to note that the commentary here is not quite correct. The least common multiple of a list of primes is only their product if the terms in the list are distinct. Perhaps the terminology implies this, but I see no direct reason to assume so.
@PauloCereda I don't actually have an HTML output module at present! It currently spits out some sort of Markdown derivative (you can choose between several).
@AndrewStacey For Euclid the "multitudes" are finite. Of course the lcm is the product of the primes, but it can easily be interpreted as a segment, while the product can't (without useless tricks). I'd prefer to write "If (p=p_j), for some (j), then ...", avoiding the \exists.
@PauloCereda Hmm, I'd need to clean up my stuff before publishing it (lots of temporary files in the repository that shouldn't be in a public place!). Would you be willing to send me a sample file for me to see if it works first?
@egreg No, the lcm of (2,2,3) is 6, not 12. Pedantic, I know. As for the \exists, you should declare your variables before using them so "if there is some \(j\) such that \(p = p_j\)" would be better, but too verbose. For a presentation then actually I think that symbols are fine and the usual advice to write proper sentences should be taken with a large heap of salt.
@AndrewStacey A "multitude of primes" in Euclid's thinking consists of distinct primes. The point of using the lcm, for Euclid, is just to help geometric understanding of the matter; for a "modern" proof, the product is fine.
@NN In case you're not sure what I'm talking about, I'm thinking of my question about the contradiction symbol. As a Pedantic Mathematician, of course I am absolutely convinced that my symbol is the correct contradiction symbol and all others are mere pretenders to the throne.
@PauloCereda Close. I'd make the lines a little further apart so that the middle square is a bit larger.
@AndrewStacey Hm OK ;-) I got the idea from this answer. I came up with \newcommand{\contrad}{\mathbin{\tikz [x=1.4ex,y=1.4ex,line width=.1ex] \draw (0,0) -- (1,1) (0,1) -- (1,0) (0.3,0) -- (1.3,1) (0.3,1) -- (1.3,0);}}
This symbol looks great for an eye test. I had to double-check to make sure I had my glasses on.
To all mathematicians in the house: I was browsing our beloved Comprehensive LATEX Symbol List and found some contradiction symbols. I'm curious on why they look like lightning bolts, is because we "zap the absurd proposition"? =P
I've been nagged by a couple of people now to convert my faith to biblatexism. I tried it all in one go, and it screwed the pooch rather on my 250 page document. I'm trying it in parts with a trial document. At the moment I have two questions, the first is covered here.
There seems to be a rathe...
Biblatexism: Assuming Philipp Lehman is its Pope, @lockstep is Prefect of the Congregation of the Faith.
4
On a more serious note, the question raises an interesting issue with respect to how to get downloaded bib data into useable formats. This may be an issue for reference managers rather than TeX specifically, but I think it's definitely on topic.
I'm writing a report that needs some annoying formatting. The \chapters and \paragraphs need to be numbered using arabic numbers.
The paragraph numbering needs to be reset at the commencement of each new chapter. The \section, \subsection, and \subsubsection headings should not be numbered, nor ...
This is fairly nice, but in my opinion it needs more work. These factors have to be considered:
1. The manual is not read only by mathematicians. Thus the proof should (a) contain as little math notation as possible, and (b) contain more "intuitive" text-form statements instead (also applies for presentations in general). 2. The title "There are infinitely many prime numbers" reads easier. 3. The lcm can be dropped without any penalty in favor of a product. 4. I think it should be accented somehow that q + 1 > 1, because otherwise p.3 does not hold.
One more try…
Theorem. There are infinitely many prime numbers. Proof. 1. Assume p_1, p_2, …, p_n are the only prime numbers. 2. Let q be their product. 3. Then q + 1 is not divisible by any of them. 4. But q + 1 > 1, thus divisible by a prime number. 5. Hence there is a prime not among p_1, p_2, …, p_n.
Good points. Let me address them one-by-one. 1. (first sentence) Absolutely, which is where this needs other eyes than mine! I agree with the second sentence **in this case** but disagree with the parenthetical remark in general. In this case, the slide is not a presentation because there is not a presenter explaining what is going on. So it has to be comprehensible without explanation. In general, I think that the more concise, the better, because then the attention of the audience is distracted from the presenter for as short a time as possible.
@AndreyVihrov This is back to a "proof by contradiction" which is not Euclid's proof. But you only need to change the first sentence to "Assume p_1, p_2, ..., p_n are prime numbers".
@AndreyVihrov Me too! One never really understands a result until one has torn it apart and reproven it five different ways.
@AndreyVihrov Must admit, I have a fondness for the "Which is absurd" line, which would come in in step 3 in your version, I think. For if p_j divides q + 1 then since it also divides q, it must divide 1 - which is absurd!
@AndrewStacey I see, you take a set of primes, but don't assume it's all of them. But then you have to ensure/state there is at least one prime in the set!
It could be as non-intrusive as Let p_1, p_2, …, p_n, n >= 1, … or Let p_1, p_2, …, p_n be prime numbers, n >= 1. But it has to be somewhere definitely.
@JosephWright This answer of Audrey has become CW because it was edited 10 times. As CW, it will no longer generate reputation, nor will is upvotes count for tag badges. Could you use your moderator powers to "un-CW" the answer? (I believe "un-CWing" has been introduced recently.)
@AndreyVihrov To be honest, if I were giving this as a presentation then all those details are things that I would say but not actually write. As I frequently tell my students "I lie a lot in lectures". So I don't think that this proof has to be 100% foolproof, but just has to be such that it doesn't say anything that couldn't be refined by the presenter.
@JosephWright Do you mean that all this has been for nothing!!!!
@JosephWright Thanks -- and good to know "un-CWing" is indeed possible. (One answer of mine with 20 upvotes has been edited 9 times, and I was dreading the 10th edit.)
@JosephWright Okay, but as a thought exercise it's been quite fun. Since you're here, what parameters would you put on the proof? How close should it be to Euclid's original?
@AndrewStacey I just noticed that in the "don't consider all" version there's something more to be proven. Namely, suppose we pick S = { a } and prove there is p != a. On another day, we pick S = { b } and prove there is q != b. But what if p = b and q = a and they are the only primes?
@AndrewStacey If this goes on, we are only able to prove there is one more prime than in a given finite set on each iteration. To prove that there is infinitely many of them, we would have to repeat the proof an infinite number of times!
Here's a candidate for "close as too localised": tex.stackexchange.com/q/27874/86 Note that the OP hasn't been back since asking the question. (2 votes at time of writing, needs 3 more.)
@AndrewStacey The question is only one week old, but I voted for "too localized" because the OP's last comment reveals that the problem was an old package version.
@AndreyVihrov That's why we do the whole lot in one go. The proof says "If P is a finite set of primes then P does not generate N". (Your objection is reminiscent of a common objection to Cantor's proof of the uncountability of the reals.)
@lockstep Exactly. The comments show that this either should be quickly answered or closed as "too localised".
@JosephWright As long as the new proof isn't in the form of a \href{http://en.wikipedia.org/wiki/Euclid%27s_theorem}{Click here to see Euclid's proof}. =P
I want all of my \subsection behave like \subsection* Instead of adding * everywhere, I like to add a few lines in the preamble. What should I do? Can I just \renewcommand?
TikZperts: is there a way to use absolute positioning (e.g., (1,0) ) with the automata library instead of the relative positioning (e.g, below right=)?
aaaah found it! (Holy RTFM, Batman)\node[state] (q_1) at (2,0) {$q_1$};
for the music-inclined, a musical saw concert: time.com/time/video/player/…(note to self: emulate a saw in an electric guitar by using a classic slow motion effect pedal)
@PauloCereda A famous French horn player, Dennis Brain, once played a concert by Leopold Mozart with a rubber hose pipe, trimming the hose to length with garden shears to achieve the correct tuning.
It's a favorite anecdote of a friend of mine, who is also a French horn player.
Dennis Brain was an Englishman, it's the horn which is French. :)
It reminds me of a famous guy in here, Hermeto Pascoal. I watch him live in a concert, and he's very "crazy". Quoting his Wikipedia entry, "known as o Bruxo (the Sorcerer), Hermeto often makes music with unconventional objects such as teapots, children's toys, and animals, as well as keyboards, button accordion, melodica, saxophone, guitar, flute, voice, various brass and folkloric instruments."
Hm the French horn is our "trompa", I was trying to find which instrument it was.
So, I now coded a package to get the text area of the current page as a special PGF/TikZ node like the current page one. I used code like that already for some answers here but just now found time to put it in a package.
I want all of my \subsection behave like \subsection* Instead of adding * everywhere, I like to add a few lines in the preamble. What should I do? Can I just \renewcommand?
