Author Archive

(or How I implemented an unanswered question tracker and began to grasp the size of the site.)

I’m not sure when it happened, but Math.StackExchange is huge. I remember a distant time when you could, if you really wanted to, read all the traffic on the site. I wouldn’t recommend trying that anymore.

I didn’t realize just how vast MSE had become until I revisited a chatroom dedicated to answering old, unanswered questions, The Crusade of Answers. The users there especially like finding old, secretly answered questions that are still on the unanswered queue — either because the answers were never upvoted, or because the answer occurred in the comments. Why do they do this? You’d have to ask them.

But I think they might do it to reduce clutter.

Sometimes, I want to write a good answer to a good question (usually as a thesis-writing deterrence strategy).

And when I want to write a good answer to a good question, I often turn to the unanswered queue. Writing good answers to already-well-answered questions is, well, duplicated effort. [Worse is writing a good answer to a duplicate question, which is really duplicated effort. The worst is when the older version has the same, maybe an even better answer]. The front page passes by so quickly and so much is answered by really fast gunslingers. But the unanswered queue doesn’t have that problem, and might even lead to me learning something along the way.

In this way, reducing clutter might help Optimize for Pearls, not Sand. [As an aside, having a reasonable, hackable math search engine would also help. It would be downright amazing]

And so I found myself back in The Crusade of Answers chat, reading others’ progress on answering and eliminating the unanswered queue. I thought to myself How many unanswered questions are asked each day? So I wrote a script that updates and ultimately feeds The Crusade of Answers with the number of unanswered questions that day, and the change from the previous day at around 6pm Eastern US time each day.

I had no idea that 166 more questions were asked than answered on a given day. There are only four sites on the StackExchange network that get 166 questions per day (SO, MathSE, AskUbuntu, and SU, in order from big to small). Just how big are we getting? The rest of this post is all about trying to understand some of our growth through statistics and pretty pictures. See everything else below the fold.

more »

While teaching a largely student-discovery style elementary number theory course to high schoolers at the Summer@Brown program, we were looking for instructive but interesting problems to challenge our students. By we, I mean Alex Walker, my academic little brother, and me, David Lowry-Duda. After a bit of experimentation with generators and orders, we stumbled across a proof of Wilson’s Theorem, different than the standard proof.

Wilson’s theorem is a classic result of elementary number theory, and is used in some elementary texts to prove Fermat’s Little Theorem, or to introduce primality testing algorithms that give no hint of the factorization.

Theorem 1 (Wilson’s Theorem) For a prime number \({p}\), we have $$ (p-1)! \equiv -1 \pmod p. \tag{1}$$

The theorem is clear for \({p = 2}\), so we only consider proofs for “odd primes \({p}\).”

The standard proof of Wilson’s Theorem included in almost every elementary number theory text starts with the factorial \({(p-1)!}\), the product of all the units mod \({p}\). Then as the only elements which are their own inverses are \({\pm 1}\) (as \({x^2 \equiv 1 \pmod p \iff p \mid (x^2 – 1) \iff p\mid x+1}\) or \({p \mid x-1}\)), every element in the factorial multiples with its inverse to give \({1}\), except for \({-1}\). Thus \({(p-1)! \equiv -1 \pmod p.} \diamondsuit\)

Now we present a different proof.

Take a primitive root \({g}\) of the unit group \({(\mathbb{Z}/p\mathbb{Z})^\times}\), so that each number \({1, \ldots, p-1}\) appears exactly once in \({g, g^2, \ldots, g^{p-1}}\). Recalling that \({1 + 2 + \ldots + n = \frac{n(n+1)}{2}}\) (a great example of classical pattern recognition in an elementary number theory class), we see that multiplying these together gives \({(p-1)!}\) on the one hand, and \({g^{(p-1)p/2}}\) on the other.

As \({g^{(p-1)/2}}\) is a solution to \({x^2 \equiv 1 \pmod p}\), and it is not \({1}\) since \({g}\) is a generator and thus has order \({p-1}\). So \({g^{(p-1)/2} \equiv -1 \pmod p}\), and raising \({-1}\) to an odd power yields \({-1}\), completing the proof \(\diamondsuit\).

After posting this, we have since seen that this proof is suggested in a problem in Ireland and Rosen’s extremely good number theory book. But it was pleasant to see it come up naturally, and it’s nice to suggest to our students that you can stumble across proofs.

It may be interesting to question why \({x^2 \equiv 1 \pmod p \iff x \equiv \pm 1 \pmod p}\) appears in a fundamental way in both proofs.

This post appears on the author’s personal website davidlowryduda.com and on the Math.Stackexchange Community Blog math.blogoverflow.com.

Welcome to the Math StackExchange community blog!

Four years ago today, Dan Dumitru proposed the creation of Math.StackExchange. The site has evolved into a community with hundreds of thousands of questions and answers, and over 30 thousand questions and answers appearing each month.

Now, we have a blog.

This blog provides a way of going beyond the Q&A format to allow exposition and discussion. There might be posts about mathematics, or Math.SE, or a book review, or whatever seems appropriate. Anyone can contribute to the blog. This blog is written and edited by community members, and we are actively soliticing both one-time and regular contributors! So if you have an idea of something you’d like to hear about or read about, or if you would like to contribute, check out the blog chat room and the Blog FAQ thread on meta.

I’m looking forward to see what we make here.