This morning on Twitter, This Day in Math posted this:

. . . it made me wonder about two things:

1. How many other palindromic primes are there between 1 and 365?

2. What in the world in a palindromic prime?

(I know these questions probably seem out of order, but what can I say . . . that’s the order I thought of them!)

Even though I thought of the questions in backwards order, clearly we need to figure out what a palindromic prime is before we can figure out the number of palindromic primes between 1 and 365.

Do you know what a palindrome is? Its a word that spells the same thing forward and backward. For example my sister’s name, Anna is a palindrome! And, if you’re a reader of this blog you certainly know what a prime number is (if you want to read more about the primes on this blog check here and here).

Given the definitions of palindrome and prime, one could reasonably assume that a palindromic prime is a prime number whose digits are the same when written forwards or backwards (and one would be correct).

Here’s the question I was left with though . . . Is a single digit number a palindrome? (And as long as we’re on the topic, is a single letter word a palindrome?–Believe it or not, someone wrote an article about this). According to Google, the answer is that single digit numbers or single letter words are palindromes, but people don’t usually talk about them because they aren’t interesting.

For the purpose of counting the number of palindromic primes between 1 and 365, however, I am going to consider single digit prime numbers to be palindromic primes (if this offends you, just cross these numbers off of your list . . . there are only 4 of them for goodness sake!)

Anyway, palindromic primes between 1 and 365. It seems to me that first it would make sense to list all of the prime numbers between 1 and 365. (We’ve talked about slick ways to test for prime-ness here). So, using the methods we already know list away!

Then, look for the palindromes in your list of primes.

(I’ve starting a short list for you. Here’s what I have 2, 3, 5, 7, 11–This means Jan. 2, Jan. 3, Jan. 5, Jan. 7, Jan. 11 are all palindromic prime-numbered days in any given calendar year)

How many did you find and what are the corresponding dates?

You know, this Twitter post, led me to think about another question. My favorite type of named numbers are perfect numbers. A perfect number is a number that equals the sum of all of its factors (not including the number itself). For example 6 is a perfect number, because the factors of 6 are 1, 2, 3, and 6 and 1+2+3 = 6.

So, how many perfect numbers are there between 1 and 365? And what are the corresponding dates in a year?

*As an side note . . . my son’s due date was 06/28/06 and I was really excited! Can you guess why?*

Happy Counting!