Why are there no palindromic primes in the interval [1000, 2000]?
We see that 1000 and 2000 are composite. Any possible palindromic primes in this interval would have to have the form 1xx1, due to symmetry. This leaves us with 10 values for the digit that x could take on. None of these values yield a palindromic prime of the form 1xx1. For each of these values, there exists a prime factorization.
But what if we represented numbers in this interval using a different base r (the radix)? Intuitively, using base 16 would suddenly kick this interval into the three digit representations for which we did find palindromic primes in base 10. So there is the possibility that base 16 will yield palindromic primes in the interval [1000, 2000].
For example, 1061 is prime, and it is represented as 425 in base 16 (0x425). 0x425 is not a palindrome. Are there palindromic primes in the interval [0x3E8, 0x7D0]?
The following list of palindromes in base 16 was determined with a little JavaScript, starting with a generated list of primes in base 10 in the interval [1000, 2000].
0x515
0x565
0x595
0x5d5
0x727
0x737
0x757
Now, we are asked to consider whether we can find palindromes in any arbitrary base. Well, if we can choose an arbitrary base, then we can "force" a prime to become a palindrome. The number 1259 in base 1260 would have a single digit, which is trivially a palindrome. If this is difficult to visualize, consider the number 15 written in base 16, which is 0xf.