The Amazing Number 23

  1. first of all and most importantly, the number 23 is the date of my birthday: December 23, 1975. that is why this page exists. i was born one day after Srinivasa Ramanujan (22 Dec 1887), and one day before Charles Hermite (24 Dec 1822), modulo one year. if you know of a reasonably-semi-famous mathematician born on the same day as me, let me know! (perhaps this spot is being reserved for me?)
  2. 23 is a prime number.
  3. 23 and 239 are the only known numbers that require nine cubes when written as a sum of cubes. as far as i know, all other numbers can be written as a sum of no more than 8 cubes. interestingly, 239 also appears in the formula pi/4 = 4 arctan(1/5) - arctan(1/239).
  4. 23 is the solution to the so-called birthday problem (no, not to guess when my birthday is, I mean the other birthday problem): how many people must be in a group before the probability that any two of them have the same birthday is at least one-half?
  5. a very similar-sounding but actually quite different problem is: how many people must be in a group with me before the probability that one of them shares my birthday is at least one-half? answer: 253. why is this significant? because it is the number of pairs that can be made with 23 people! which naturally enters into the solution of the previous problem. Quelle coincidence!
  6. the first cyclotomic field to have non-unique factorization is that formed by adjoining a 23rd root of unity.
  7. -23 is the smallest (in absolute value) cubic discriminant, i think. it is the discriminant of x^3-x-1 or something like that. this polynomial is also significant because it has the smallest mahler measure among all non-reciprocal polynomials (which should be called palindromials in my opinion) (smyth's theorem). either of these statements might be slightly wrong; someone please correct me if they are. more about mahler measure
23 is not actually my favorite number. but so much is already written about 11, 13 and 19, that i thought it would be best to talk about 23 instead.
13 is my "lucky number" while 11 and 19 have spiritual significance for me.
compared to other prime numbers of similar magnitude, i find 17 uninteresting. and let us not even talk about composite numbers!

All of these amazing facts about the number 23 (except the first and most amazing) can be found at Prime Curios, but there's also so much junk there that it's hard to find what's interesting. I mean, consider this entry:

23 is the smallest prime of the form 10*p + 3 that is not the sum of two squares, where p is prime.
Who contributes this stuff? And who cares how many counties are in Maryland?

thanks for coming by. i leave you with this little rhyme (not written by me!):

Thrice three and twenty for pleasure true and plenty.