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 
, 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 
is prime (of course really a number is either prime or it’s not) is 
(
is the log base e, e being the constant more important than 
equal to around 2.718. The log base b of n, or 
if 
, so it is the opposite of taking 
to 
.) So the “chance” of a number around 2000 being prime is roughly 
. Once again I have underestimated how slowly the log function grows, because 
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 being prime and 
being prime aren’t independent events so you can’t multiply their probabilities!)
3. 2011 can be written as the sum of 3 squares: 
. 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 
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 
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Posted by Luyi on January 2, 2011 at 5:11 pm
Hi Meena! This post is excellent, very creatively informative. I love your blog! Happy 2011
Posted by meenamathgirl on January 2, 2011 at 5:47 pm
Thanks Luyi!! Happy new year.
Posted by Anonymous on January 2, 2011 at 8:01 pm
Wolfram Alpha!
Posted by anon on January 2, 2011 at 8:01 pm
Wolfram Alpha!
Posted by CatSynth » 2011 on January 14, 2011 at 2:06 pm
[...] The Power of Proofs, we have the property that 2011 is the sum of three [...]