In working out prime birthdays, James Grime remarked that his 1331th prime birthday would be “extra special”, and left it to his followers to work out why. The answer, of course, is that it is also his annual birthday. This is an interesting phenomenon: when he is 10957 days old, James Grime will also be 30 years old. This is obviously not true for everyone: there are 365.25 days per year. We count 365 in a normal year and once every four years we catch up by adding an extra day, 29th Feb, in a leap year. Now, 365.25*30 = 10957.5, so the fact that James’ 30th birthday coincides is not going to be true for everyone. What are the conditions in which this happens?

Well, whether this happens or not is related to how many catch-up leap years you’ve crossed during your 10957 days. James was born in 1980, in October. Being born in the first few months after a 29th Feb, James doesn’t benefit from his first catch-up for almost four years. This means he is generally owed, rather than owing a day. If he had been born in the months running up to a 29th Feb, he would experience his first extra day very quickly and generally be ‘ahead’.

Anyway, further questions occur: how many times might this occur in your life? And where in the leap year cycle would you need to be born in order to benefit from this phenomenon on that birthday? Here is a list. I could explain how I came about this, but this margin is too narrow. It has to do with where you are born in relation to the leap day and whether the prime is above or below 365.25 * number of years.

First, some terminology. Let y_i be a period of 365 days from 1st March to 28th Feb. Let d be the 29th Feb. Then the leap year cycle looks like this:

y₁y₂y₃y₄d

Then the prime birthdays that are also annual birthdays look like this:

I may very easily have calculated, or typed, any of this completely incorrectly.