2011: An Interesting Number?

Yesterday there was a fair bit of comment on Twitter about how 2011 is the sum of 11 consecutive prime numbers.

Which was nice.

The entrance doors to Kirkconnel Parish Church

The entrance doors to Kirkconnel Parish Church

? Copyright Walter Baxter and licensed for reuse under this Creative Commons Licence

But is this the only reason why 2011 is an interesting number? Hell, no. Here are a few more of the qualities that 2011 possesses.

  1. It is a prime number, n such that 2n-1 is also prime. So if you double 2011 and take one away you get 4021, which is also a prime number. The last time this happened was in 1867.
  2. It is a toothpick prime. Toothpick numbers are created by imagining that you are putting toothpicks on a table according to certain rules. Here is a really good simulation of toothpick sequences, they display some lovely symmetries and look a bit fractally at times. 1871 was the last year that was a toothpick prime. The next one will be in 2731.
  3. It is a number that eventually reaches 1 under “x -> sum of cubes of digits of x”. So, you cube each of the digits of a number and add them together, then repeat with the result you obtain. Now, most numbers immediately start to increase quite dramatically under this rule, but 2011 becomes 2^3 + 0 + 1 + 1 = 8+1+1 = 10 and 10 becomes 1. The last year this was true was 1981 and it won’t happen again until 2101
  4. It is a prime, p such that p+1 is divisible by each digit in p. So, if you add 1 to 2011 you get 2012 which is divisible by 0,1 and 2. Now, this doesn’t look too special at first, but although it last happened in 2003 it won’t be true again until 2063. So, savour it while you can

And there are loads more qualities that 2011 has. If you’ve made it this far down then you probably won’t be surprised that you can find an interesting quality or two about nearly any number. In order to find the four above I searched the Online Encyclopaedia of Integer Sequences, where 2011 is listed in 401 different sequences. In fact, you have to get to the number 11630 until you reach a number not listed in any sequence.

So, does this mean that we will have to wait until the year 11630 to have an uninteresting number?

Well no, because mathematicians have found a nice little paradox. The Interesting Number Paradox to be precise, which states that all numbers are interesting. Even if they aren’t.

It runs like this. You can define what is interesting or uninteresting however you like. Once you have done this you have a set of interesting numbers and a set of uninteresting numbers. In the uninteresting set there is one number which is the lowest value. Hey presto, you’ve just found something interesting about it!

Now, bung that number in the interesting set and go back to the uninteresting set. There’s a new value which is now interesting by virtue of being the lowest uninteresting number. And in this way you can move up through the uninteresting number set until all of them are interesting.

So, every new year we should expect to have have a round of Tweets expressing at least one wonderful property of the year’s value. I for one look forward to that.


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