Numerological facts of 2015

Did you wake up on January 1, 2015, and wonder what 2015 is about? Me too. And it turns out that MMXV has some fun patterns hiding in it.The first thing that I found out about 2015 is that it is a Lucas-Carmichael number, which according to Wikipedia means that it “is a positive composite integer n such that if p is a prime factor of n, then p + 1 is a factor of n + 1″. Concretely, the prime factors of 2015 are 13, 5, 31; so n=2015 and p=13, 5, 31. That means that 14, 6, 32 are all factors of 2016.

Well, now what? 2015 isn’t prime – we have to wait till 2017 for another one of those. It can’t be written as the sum of three squares – but it can be written as a sum of four squares. Maybe it’s interesting that the closest twin primes are 1997 and 1999? I was worried too until I saw that 2015 can be written as 2^11 – 33 which itself can be rewritten as 2^11 – 2^5 – 1. What if we switched bases? In binary, 2015 is written 11111011111. I don’t know about you, but I love palindromes (did you notice the prime factors palindrome?). What about other basses? In base 4, it would be written as 133133; base 8: 3737; base 64: 3131. Aha! Glorious pattern!

Unfortunately 2015 written base 16 and base 32 doesn’t contain much pattern. But just for fun, let’s see what it looks like in base 16: 7df. Any colorists around? That’s a rather blue looking number to me. Maybe it would help if I rewrote it as #7df. And for one final trick, let’s write the IEEE floating point representation of 2015. 000…07c9f40 those last 3 bytes make a rather green color.


If you can think of other interesting facts about 2015, please let us know! I hope you have a very blue-green, palindromic year!
Numerological facts of 2015