A Curious Coincidence

My first wife, and mother of my daughter, was born on the 23rd November 1953 and will be turning 63 shortly. My daughter was born on the same date in 1980 and will be turning 36. I've written about the mathematical aspects of this digit reversal in an earlier post. 

Getting back to my first wife however, she will be turning 23011 days old. A curious coincidence indeed. Furthermore, the number is prime and marks the start of five consecutive primes: 23011, 23017, 23021, 23027 and 23029. It is also the first member of a prime quadruple in a 2p-1 progression: 23011, 46021, 92041 and 184081. This is a Cunningham chain of the second kind. I've written about such chains in an earlier post.

However, I'm interested in the general question of what are the conditions for such coincidences to occur? It can be noted that 23011/365.242199 is about 63.00203 so such coincidences can only occur when the day count is almost exactly divisible by 365.242199. Now a person must be born between the 1st of January and the 31st December, thus the possible numbers generated (using leading zeroes) range from 01001 to 31012. 

On a spreadsheet, I tested division over every number in this range by 365.242199, looking for remainders that were less 0.01. A divergence of just two days either side produces a difference of around 0.005 so 0.01 is quite generous. A few pockets of numbers satisfied the condition but only 23011 held up in the end. 23012, corresponding to 23rd December, came close but birthdays fell on 23rd November or 23rd October so it didn't satisfy.

Even having this birthday does not guarantee the coincidence. For example, my daughter will turn 63 in 2043 but on that day she will be 23010 days old. While on the topic of coincidences, I must note that on the 23rd November 2016, I will be 24706 days old. This number factorises to 2 x 11 x 1123. Amazing.