There is only one palindromic prime with an even number of digits.
Prove it.
If a palindrome has an even number of digits, then the sum of the odd digits = the sum of the even digits. This means that it equals 0 mod 11, because 10^n is 1 mod 11 if n is even and -1 mod 11 if n is odd.
For instance 3223 mod 11 =
(3 * 1000 mod 11) + (2 * 100 mod 11) + (2 * 10 mod 11) + 3 =
3*(-1) + 2*1 + 2*(-1) + 3*1 = 5*(-1) + 5*1 = 0
Because all palindromes with even digits are divisible by 11, the only one which can be prime is 11.
Edited on December 28, 2010, 6:06 pm
Edited on December 29, 2010, 12:35 am
Solution (spoiler)
