# Rambling Thoughts

## 2012

Today marks the start of a new year: 2012. I’d like to take this opportunity to say a few things about this number.

• $2012 = 2 \cdot 2 \cdot 503$
• $2012, 2013, 2014, 2015$ have the same number of primes in their prime factorizations (counting multiplicities). The smallest numbers $n$ such that $n, n+1, n+2, n+3$ have the same number of primes in their prime factorizations are $602, 603, 1083, 2012$, so this is actually fairly rare.

And some more really rare properties:

• 2012 is not odd.
• 2012 is not prime.
• 2012 is not a square.
• 2012 is not squarefree.
• 2012 is not a perfect number.

I know I haven’t posted much recently. It’s a poor excuse, but I’ve been busy. I do have a number of things that I want to write about, however, and I’ll get to them eventually.

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## 2011: Numerology

It is time to celebrate the first day of the new year, and I hereby welcome you to the year 2011.

Of course, the timing of the start of the year is an artifact of our calendar system. The choice of the date to be called “January 1” was determined more or less arbitrarily, and in fact, other cultures use different calendar systems with the new year occurring on different dates. As far as I can tell, time passes uniformly and there is no canonical time that deserves to be called the start of the year. In that sense, celebrating the new year is no more sensible than numerology.

In fact, 2011 (like every other number) has interesting numerological properties; given sufficient time and effort, the number of numerological properties are unbounded. Here’s a small sample:

• 2011 is prime
• 2011 is the sum of 11 consecutive primes: 2011 = 157 + 163 + 167 + 173 + 179 + 181 + 191 + 193 + 197 + 199 + 211
• The past few prime years were 1997, 1999, 2003, 2011. Notice that the successive differences of these years are consecutive powers of two. In fact, 2027 is also a prime year, though 2059 = 29 * 71 is not.
• 2011^2 = 4044121. Reversing the digits on each side, we get 1102^2 = 1214404.
• 2011 = 1 + 2 * 3 * (4 * (5 * 6 + 7 * 8) – 9)