In the year 75015151057

Yesterday Sally cleverly observed that 2015 is a palindrome in binary. Which got me wondering whether there could be any years that are palindromes in both binary and decimal.

So I wrote a little javascript program to search for them. After several hours of crunching, my computer found 36 such years, from the year one through the year 75015151057, which in binary is:

 

1000101110111010000000101110111010001

 

The most recent year on the list was 717 AD, and the next one won’t be until 7447 AD, so don’t hold your breath.

Are there infinitely many such years, or is there a largest one? Somebody would need to come up with a mathematical proof, but I can make a guess just from looking at the numbers.

As I look down the list, the number of digits is increasing at a pretty constant rate. That’s what mathematicians call a logarithmic distribution. If that pattern continues, then the sequence will go on forever.

If that’s true, then no matter how far out in the future you go, there will always be another year that is a palindrome in both base two and base ten. Although you probably won’t be able to get to most of them without a Tardis.

2 thoughts on “In the year 75015151057”

  1. Oh that’s quite all right Sally. You did cleverly observe it. You just cleverly observed it in a post somewhere. 😉

    I may be the first person to have pointed out that 2015 is a palindrome in base 64. I don’t know whether that makes me clever or just weird.

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