Carnival of Mathematics 159
Friday, July 6, 2018 · 4 min read
Welcome to this month’s edition of the Aperiodical Carnival of Mathematics! The Carnival is a monthly roundup of exciting mathematical blog posts. Last month, it was hosted by Paul at the Aperiodical. This month, it is my honor to host it here at Comfortably Numbered. But first…
Let’s play a game, shall we?
Pick a number. Not too large, though! You’re about to do some quick math on it. (I’ll play along with 4.)
Okay. Ready? Good.
Now take your number and square it. (4 squared is 16.) Then, add your original number to the square. (16 plus 4 is 20.) Finally, add forty-one. (20 plus 41 is 61.)
And now – your result – is it prime? Ha! I thought so. (61 certainly is.)
This little trick is due to Euler, who pointed out in 1772 that the polynomial ($ f(x) = x^2 + x + 41$) returns prime numbers for small integers — indeed, all nonnegative integers up to and including 39. Since then, the quest for other such “prime-generating” polynomials has fascinated number theorists from around the world. As a little exercise, you may try convincing yourself that there is no perfect prime-generating polynomial; that is, that there will always be at least one integer input that gives a composite output.
But I digress. Here is what matters: The integers ($ x $) for which ($ f(x) $) is composite are the deviants, the rebels, the ones who refuse to play along with Euler’s little game.
Forty is the first such integer.
One hundred fifty-nine is another.
Welcome to the 159th Carnival of Mathematics.
The Carnival always has a special place in its heart for clever ways to teach children various math concepts. And this one’s no exception. In Set Theory for Second Grade, Manan talks about how he designed an engaging lesson on set theory (and common multiples!) for second graders. A quote from his students: “Can you hang this in the hall so that everyone can see the college math we did?”
What an amazing moment — new symbols, new concepts, no problems! At this point, I made sure to remind them that what they are learning right now is no different from what I would teach in college. And that if today, here in second grade they could do college math, then in third grade they can do third grade math, in fourth grade they can do fourth grade math, and that they can always do math! More than a few students’ faces lit up.
In British Mathematical Colloquium, days 3 and 4, Peter Cameron recounts in excellent detail the last couple days of the Colloquium (it reads like a mini-Carnival!). Days 1 and 2 are linked within.
An induced subgraph of a graph is obtained by throwing away some vertices and the edges incident with them; you are not allowed to throw away an edge within the set of vertices you are keeping. Paul began with the general problem: given a graph H, can you determine the structure of graphs G containing no induced copy of H? … The answer is known in embarrassingly few cases … Not even for a 4-cycle is the answer known!
In Sum of Cubes is Square of Sum… And More!, Pat Ballew begins with a fact that most high-schoolers are taught, and then rather suddenly finds himself deep in a fascinating rabbit hole. (Editor’s note: I encourage you to read the author’s On This Day in Math series; I would list all thirty of the past month if I could…)
Like many teachers at the upper level high school math classes, over the years I’ve presented the sum of the Cubes of the natural numbers formula above many dozens of times. Then, perhaps like many others, I would point out how nice it is that it turns out to be the square of the nth triangular number, a happy coincidence that would make it easier to remember. Usually then, we would challenge them to extend the idea to fourth powers and see if they could do the induction proof, even though there was no really nice simplification (to my knowledge) of the sums of fourth powers.
But then I reread a book that has been in my library for about six years, and realized that many of those teachers may have known a different approach to sums of cubes equaling square of sums that I had been completely unaware of. In case there are other teachers who somehow also didn’t know, I share my newfound ancient knowledge.
In Thinking about the Law of Quadratic Reciprocity, Dan McQuillan gives a fast-paced overview of one of my personal favorite theorems in number theory.
Mathematics, the way it is currently written, can be difficult to read. Sometimes it helps to see how people think about a topic or theorem before (or after, or during) the reading of a proper treatment or rigorous proof. The purpose of this post is to provide such a view regarding the proof of the famous law of quadratic reciprocity. There are many details missing, on purpose, and the hope is that it reads like a good story that’s both interesting, believable and easily verifiable.
Just for fun!
In Magical Learning, John Cook reports the results from an informal Twitter poll he conducted: “If a genie offered to give you a thorough understanding of one theorem, what theorem would you choose?”
In case you missed it, Christian at the Aperiodical is running The Big Internet Math-Off. It may not have the intense moment-by-moment drama of the World Cup, but the daily tidbits of math are definitely worth subscribing for.
In Math Explained through Anagrams, Ben Orlin constructs a frankly impressive amount of anagrams for various parts of math. And they’re all illustrated! (Editor’s note: I also enjoyed this piece by the same author. I don’t want to spoil it, but here is a wonderful quotation: “I sometimes think that there are no puddles in math;” says Orlin, “there are only oceans in disguise.”)
That’s all I have for you this month! Come back next time when Robin at Theorem of the Day will host the 160th Carnival of Mathematics!