This will never happen again…

Here’s a quick, fun tidbit about dates. I was thinking about the factorization of 2013 and the next few years, and I noticed something cool. (Note this is good for psyching out math contests, which often have problems with the current year in them.)

2013 = 3 * 11 * 61

2014 = 2 * 19 * 53

2015 = 5 * 13 * 31

Each date factors into primes as (single digit) * (teen) * (double digit bigger than teen). (I’m counting 11 as a teen, i.e. I really mean a prime between 11 and 19.) So I wondered how often you get three such numbers in a row, and I wrote a little Matlab code. It turns out that these three are the only example! In fact there are only three other consecutive pairs: 1221,1222; 1598,1599; and 2821,2822.

I can’t imagine that this kind of factorization has any real importance, but I thought it was cool. The next year, 2016, is quite different, but also cool: it is

2016 = 32 * 63 = 2^5 * 3^2 * 7

which is (power of two) * (power of two – 1), and which is also “7-smooth”, meaning that it has no prime factor larger than 7. (These are sometimes called “highly composite”, but that has a different, and more interesting, meaning, about which I made some videos last year.) The 7-smooth numbers are pretty sparse; the only ones occuring in anyone’s lifetime who is alive right now are 1920, 1944, 1960, 2000, 2016, 2025, 2048, 2058, 2100. Note that 2048 = 2^11, which will be cause for worldwide, year-long celebration, no doubt. 🙂

Speaking of numbers of the form (power of two) * (power of two – 1), if a number n is n = 2^(k-1) * (2^k – 1), and (2^k – 1) is prime (a Mersenne prime), then n is a perfect number. Now 2016 doesn’t fall into that category, since 63 is not prime, but it is a number with a particularly large sum of all of its factors, called sigma(n) (which, for a perfect number n, would equal exactly 2n). The sum of factors of 2016 is sigma(2016) = 6552, which is higher than for any other number in the years 1900-2099 except for 1980 (which ties it with sigma(1980) = 6552). The year 2040 is very close, with sigma(2040) = 6480, and 2100 beats it with sigma(2100) = 6944.

The story is very similar if you just look at the number of divisors (often called sigma_0(n) or tau(n)). The number n = 2016 again beats everything except 1980 and 2100, which both tie it with tau(n) = 36. So in 2016, let’s make sure to do a lot of division!

If you watch my videos about highly composite numbers (“Why 5040 is cool”) you’ll see that numbers that factor into small primes (but not all 2’s, like 2048, or 2’s and 3’s) tend to have the largest number of divisors and the highest sigma values.

Going one more year out, 2017 is prime. Going back a little, 2012 = 2 * 2 * 503, and 2011 was prime. (Thanks to oeis.org for a lot of the data for all of these statements!)

Hope that was fun.

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One more quick one: musical geometry and topology

Just have a look at the inimitable George Hart’s video. I particularly love the connection between sculpture, music, and math at the end.

And in a similar vein, you really should take a look at ur-blogger John Baez’s new blog for the AMS, Visual Insight, dedicated to amazing mathematical visualizations.

Enjoy!

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Getting students thinking right from the start

Here’s a little idea I just thought of, that I’d like to try. I’m always thinking about ways to get students to ask “does this make sense?” “Is this answer reasonable?” Well, what about putting that into the statement of the question? So for example, take this typical calculus problem:

A plane flies horizontally over a radar station at an altitude of 1km at a speed of 500 km/hr. How fast is the distance to the radar station changing when the plane is 2km away from the radar station and heading away from it?

…and replace it with this one:

A plane flies horizontally over a radar station at an altitude of either 1.00m, 1.00km, or 1000km (whichever makes sense) at a speed of either 5.00 km/hr, 500 km/hr, or 50,000 km/hr (whichever makes sense). How fast is the distance to the radar station changing when the plane is either 2.00m, 2.00km, or 20,000km away from the radar station (whichever makes sense) and heading away from it? Before doing any calculation, (a) what units will the answer have, (b) will it be positive, negative, or zero, and (c) is it likely to be in the range [-1000,-500],[-500,0],[0,500],[500,1000], or none of the above? Justify your reasoning. Then analyze the problem carefully and give a numerical answer, to appropriate precision.

A more creative option than the “multiple-choice question” format above would be to have every student (or every group) make up their own problem, making sure that they can justify the reasonableness of the numbers they put in. But that might be unwieldy, with many different answers that would be hard to check and hard to have a whole-class discussion about.

We’ll see if I can find time to put this into practice.

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More progress on prime gaps

Courtesy of Not Even Wrong, we have news of further progress on the size of prime gaps. The polymath8 project is readying its paper that proves that gaps no bigger than 4680 occur infinitely often, but apparently they’ve been scooped by James Maynard, who has a paper on the arXiv preprint server that brings that number down to 600. There’s a very good expository article in the Simons Foundation‘s Quanta Magazine about the whole story.

If the number 600 holds up, I think that’s a nice improvement, since gaps of size 600 start to show up (as unusually large gaps, admitted) for numbers in the trillions, which are rather small numbers by the standards of number theory or, say, cryptography. And a gap of 600 becomes average for numbers with around 250 digits, well within the usual bounds of cryptography. So one could definitely say that they have proved that “the typical size gap for practical applications of prime numbers recurs infinitely often.” Not the twin prime conjecture, to be sure, but very nice.

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Calculus and the Holy Grail

I just posted this video on YouTube mixing algebra, calculus, and Medieval Errantry. Will the good knight escape his castle prison and vanquish the evil knight? Watch and find out!

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A World of Waves: video series

I haven’t added it yet to “video links”, but I just posted a video series on YouTube about Fourier series and how they have impacted modern science and technology and especially modern mathematics. It’s called “A World of Waves“. It’s very non-technical, so if you want the calculational details, it’s not for you, but if you want some interesting context and history, try it out.

Update: it’s now included in the “Advanced Math for a General Audience” page under “Video links”.

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A fun question about handedness

Here’s a very quick question, which I actually got from OK Cupid, of all places: if you turn a left-handed glove inside-out, does it (a) stay a left-handed glove, (b) turn into a right-handed glove, or (c) become neither? Think about it, and stay tuned for an answer and discussion of such issues very soon.

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