A friend made me aware of a neat little unsolved problem in number theory. I know it seems like number theory is nothing but unsolved problems, but this is an unfair reputation. There are as many as four solved problems in number theory. It’s a tough field.
The question started with the observation that 11 is a prime number. And so is 101. But 1,001 is not; nor is 10,001. How many prime numbers are there that have the form , for whole-number values of n? Are there infinitely many? Finitely many? If there’s finitely many, how many are there?
It turns out this is an open question. We know of three prime numbers that you can write as . I’ll leave the third for you to find.
One neat bit is that if there are more prime numbers, they have to be ones where n is itself a whole power of 2. That is, where the number is for some whole number k. They’ve been tested up to at least, so this subset of the Generalized Fermat Numbers seems to be rare. But wouldn’t it be just our luck if from onward they were nothing but primes?
Greetings, friends, and thank you for visiting the 136th installment of Denise Gaskins’s Playful Math Education Blog Carnival. I apologize ahead of time that this will not be the merriest of carnivals. It has not been the merriest of months, even with it hosting Pi Day at the center.
In consideration of that, let me lead with Art in the Time of Transformation by Paula Beardell Krieg. This is from the blog Playful Bookbinding and Paper Works. The post particularly reflects on the importance of creating a thing in a time of trouble. There is great beauty to find, and make, in symmetries, and rotations, and translations. Simple polygons patterned by simple rules can be accessible to anyone. Studying just how these symmetries and other traits work leads to important mathematics. Thus how Kreig’s page has recent posts with names like “Frieze Symmetry Group F7” but also to how symmetry is for five-year-olds. I am grateful to Goldenoj for the reference.
That link was brought to my attention by Iva Sallay, another longtime friend of my little writings here. She writes fun pieces about every counting number, along with recreational puzzles. And asked to share 1458 Tangrams Can Be A Pot of Gold, as an example of what fascinating things can be found in any number. This includes a tangram. Tangrams we see in recreational-mathematics puzzles based on ways that you can recombine shapes. It’s always exciting to be able to shift between arithmetic and shapes. And that leads to a video and related thread again pointed to me by goldenoj …
This video, by Mathologer on YouTube, explains a bit of number theory. Number theory is the field of asking easy questions about whole numbers, and then learning that the answers are almost impossible to find. I exaggerate, but it does often involve questions that just suppose you understand what a prime number should be. And then, as the title asks, take centuries to prove.
Fermat’s Two-Squares Theorem, discussed here, is not the famous one about . Pierre de Fermat had a lot of theorems, some of which he proved. This one is about prime numbers, though, and particularly prime numbers that are one more than a multiple of four. This means it’s sometimes called Fermat’s 4k+1 Theorem, which is the name I remember learning it under. (k is so often a shorthand for “some counting number” that people don’t bother specifying it, the way we don’t bother to say “x is an unknown number”.) The normal proofs of this we do in the courses that convince people they’re actually not mathematics majors.
What the video offers is a wonderful alternate approach. It turns key parts of the proof into geometry, into visual statements. Into sliding tiles around and noticing patterns. It’s also a great demonstration of one standard problem-solving tool. This is to look at a related, different problem that’s easier to say things about. This leads to what seems like a long path from the original question. But it’s worth it because the path involves thinking out things like “is the count of this thing odd or even”? And that’s mathematics that you can do as soon as you can understand the question.
I again thank Iva Sallay for that link, as well as this essay. Dan Meyer’s But Artichokes Aren’t Pinecones: What Do You Do With Wrong Answers? looks at the problem of students giving wrong answers. There is no avoiding giving wrong answers. A parent’s or teacher’s response to wrong answers will vary, though, and Meyer asks why that is. Meyer has some hypotheses. His example notes that he doesn’t mind a child misidentifying an artichoke as a pinecone. Not in the same way identifying the sum of 1 and 9 as 30 would. What is different about those mistakes?
Jessannwa’s Soft Start In The Intermediate Classroom looks to the teaching of older students. No muffins and cookies here. That the students might be more advanced doesn’t change the need to think of what they have energy for, and interest in. She discusses a class setup that’s meant to provide structure in ways that don’t feel so authority-driven. And ways to turn practicing mathematics problems into optimizing game play. I will admit this is a translation of the problem which would have worked well for me. But I also know that not everybody sees a game as, in part, something to play at maximum efficiency. It depends on the game, though. They’re on Twitter as @jesannwa.
These are thoughts about how anyone can start learning mathematics. What does it look like to have learned a great deal, though, to the point of becoming renowned for it? Life Through A Mathematician’s Eyes posted Australian Mathematicians in late January. It’s a dozen biographical sketches of Australian mathematicians. It also matches each to charities or other public-works organizations. They were trying to help the continent through the troubles it had even before the pandemic struck. They’re in no less need for all that we’re exhausted. The page’s author is on Twitter as @lthmath.
I have since the start of this post avoided mentioning the big mathematical holiday of March. Pi Day had the bad luck to fall on a weekend this year, and then was further hit by the Covid-19 pandemic forcing the shutdown of many schools. Iva Sallay again helped me by noting YummyMath’s activities page It’s Time To Gear Up For Pi Day. This hosts several worksheets, about the history of π and ways to calculate it, and several formulas for π. This even gets into interesting techniques like how to use continued fractions in finding a numerical value.
Rolands Rag Bag shared A Pi-Ku for Pi-Day featuring a poem written in a form I wasn’t aware anyone did. The “Pi-Ku” as named here has 3 syllables for the first time, 1 syllable in the second line, 4 syllables in the third line, 1 syllable the next line, 5 syllables after that … you see the pattern. (One of Avery’s older poems also keeps this form.) The form could, I suppose, go on to as many lines as one likes. Or at least to the 40th line, when we would need a line of zero syllables. Probably one would make up a rule to cover that.
This one I saw through John Allen Paulos’s twitter feed. He points out that it’s like the Collatz conjecture but is, in fact, proven. If you try this yourself don’t make the mistake of giving up too soon. You might figure, like start with 12. Sum the squares of its digits and you get 5, which is neither 1 nor anything in that 4-16-37-58-89-145-42-20 cycle. Not so! Square 5 and you get 25. Square those digits and add them and you get 29. Square those digits and add them and you get 40. And what comes next?
This is about a proof of Fermat’s Theorem of Sums of Two Squares. According to it, a prime number — let’s reach deep into the alphabet and call it p — can be written as the sum of two squares if and only if p is one more than a whole multiple of four. It’s a proof by using fixed point methods. This is a fun kind of proof, at least to my sense of fun. It’s an approach that’s got a clear physical interpretation. Imagine picking up a (thin) patch of bread dough, stretching it out some and maybe rotating it, and then dropping it back on the board. There’s at least one bit of dough that’s landed in the same spot it was before. Once you see this you will never be able to just roll out dough the same way. So here the proof involves setting up an operation on integers which has a fixed point, and that the fixed point makes the property true.
John D Cook, who runs a half-dozen or so mathematics-fact-of-the-day Twitter feeds, looks into calculating the volume of an egg. It involves calculus, as finding the volume of many interesting shapes does. I am surprised to learn the volume can be written out as a formula that depends on the shape of the egg. I would have bet that it couldn’t be expressed in “closed form”. This is a slightly flexible term. It’s meant to mean the thing can be written using only normal, familiar functions. However, we pretend that the inverse hyperbolic tangent is a “normal, familiar” function.
For example, there’s the surface area of an egg. This can be worked out too, again using calculus. It can’t be written even with the inverse hyperbolic cotangent, so good luck. You have to get into numerical integration if you want an answer humans can understand.
I dreamed recently that I opened the Sunday comics to find Scott Adams’s Dilbert strip turned into a somewhat lengthy, weird illustrated diatribe about how all numbers smaller than infinity were essentially the same, with the exception of the privileged number 17, which was the number of kinds of finite groups sharing some interesting property. Before I carry on I should point out that I have no reason to think that Scott Adams has any particularly crankish mathematical views, and no reason to think that he thinks much about infinity, finite groups, or the number 17. Imagining he has some fixation on them is wholly the creation of my unconscious or semiconscious mind, whatever parts of mind and body create dreams. But there are some points I can talk about from that start.