## Properties of 2011

So, it’s the new year.  I challenged myself to think of as many cool mathematical properties about 2011 and this is what I came up with…

1.  2011 is a prime number!  To verify this, you have to check that none of the prime numbers less than $\sqrt{2011}$, which rounded down is 44, divide 2011.

2.  As my friends have pointed out, 2011 is a sexy prime, since it is a member of a pair of 2 primes 6 apart, which is (2011, 2017).  Though there seems to be no such term, it’s an octy prime as well since 2003 is prime as well.  After 2017, the next prime is 2027.

That so many primes occur in the 2000’s surprises me, but it’s actually not too surprising.  The Prime Number Theorem says that the “chance” that a number $n$ is prime (of course really a number is either prime or it’s not) is $\frac{1}{\ln{n}}$ ( $\ln{n}$ is the log base e, e being the constant more important than $\pi$ equal to around 2.718.   The log base b of n, or $\log_{b} n = x$ if $b^x=n$, so it is the opposite of taking $x$ to $b^x$.)  So the “chance” of a number around 2000 being prime is roughly $\frac{1}{\ln{2000}}=\frac{1}{7}$.  Once again I have underestimated how slowly the log function grows, because $\ln{2000}$ isn’t very big.

I guess you could say then that the probability of 2011 being a sexy prime is rare, but really a sexy prime is so contrived!!  (The Wikipedia article has failed to convince me of its usefulness, at least.)  (EDIT: Finding the probability of two numbers being sexy primes is actually harder than I thought because $n$ being prime and $n+6$ being prime aren’t independent events so you can’t multiply their probabilities!)

3.  2011 can be written as the sum of 3 squares: $39^2+21^2+7^2$.  Lagrange’s Four Square Theorem says that every number can be written as the sum of 4 squares in at least 1 way.  An extension of that (Jacobi’s Four Square Theorem) says that the number of ways a number can be written as the sum of 4 squares is 8 times the sum of its divisors for odd integers, and since 2011 is prime, this is easy to find–it’s just $8 \cdot (2011+1) = 16,096$ ways.  (You can find the sum of the divisors of a number in general as well by taking the sum of the divisors of each of the prime powers in its factorization and multiplying them.)  Don’t try listing all 16,096 ways 😛

Have a great 2011 🙂

### 5 responses to this post.

1. Hi Meena! This post is excellent, very creatively informative. I love your blog! Happy 2011 🙂

2. Thanks Luyi!! Happy new year.

3. Posted by Anonymous on January 2, 2011 at 8:01 pm

Wolfram Alpha!

4. 