Some 2016 fun

It’s 2016! This is awesome. I wanted to create a repository of awesome facts on the number 2016. Anyone who has any nontrivial facts about the number can comment!

  • Obviously, we start off with \#\mathrm{GL}_2(\mathbf{Z}/7\mathbf{Z})=2016, as in our previous post.
  • Relevant to my project under Professor Elkies: it’s a congruent number!
  • 1+2+...+63=\frac{63\times 64}{2}=2016.
  • 2016={64\choose 2}. It’s the 63rd triangular number. It’s also hexagonal (because 63=2\times 32 - 1, it’s the 32nd hexagonal number).
  • Scott Kominers tells me that 2016 is also icositetragonal. So it’s expressible in the form \frac{1}{2}(22n^2 - 20n) . But honestly speaking, “icositetragonal” is a really cool name for a number. If I was a (human) icositetragonal number, I’d be a happy baby potato.
  • You can prime factorize 2016 as 2^5\times 3^2\times 7, so there are exactly fourteen abelian groups of order 2016 (where we’re utilizing the Chinese remainder theorem).
  • It’s divisible by the number of its divisors.
  • \frac{1}{10}\times \#A_8 = \frac{1}{10}\times \#\mathrm{GL}_4(\mathbf{Z}/2\mathbf{Z})/10 = 2016. The cool thing is that A_8\cong \mathrm{GL}_4(\mathbf{Z}/2\mathbf{Z}). I’d like to write up a proof of this someday.
  • From Facebook: “The 64 primes from 79 to 439 can be put into an 8×8 magic square so that each row, column and diagonal sums to 2016:

    439  89   83  97  419 379 113 397
    137 149 433 317 373 163 193 251
    331 349 199 179 313 271 223 151
    239 167 227 233 269 307 401 173
    257 353 191 263 229 103 283 337
    293 281 347 367 131 277 109 211
    241 197 127 421 101 157 383 389
    79 431 409 139 181 359 311 107″

  • Any more? Comment below!
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