A strictly non-palindromic number is an integer n that is not palindromic in any positional numeral system with a baseb in the range 2 ≤ b ≤ n − 2. For example, the number 6 is written as "110" in base 2, "20" in base 3 and "12" in base 4, none of which is a palindrome—so 6 is strictly non-palindromic. For another example, the number 19 written in base b is: None of these are a palindrome, so 19 is a strictly non-palindromic number. The sequence of strictly non-palindromic numbers starts: To testwhether a number n is strictly non-palindromic, it must be verified that n is non-palindromic in all bases up to n − 2. The reasons for this upper limit are:
any n ≥ 3 is written "11" in base n − 1, so n is palindromic in base n − 1;
any n ≥ 2 is written "10" in base n, so any n is non-palindromic in base n;
any n ≥ 1 is a single-digit number in any base b > n, so any n is palindromic in all such bases.
For example, 19 will be written as: Thus it can be seen that the upper limit of n − 2 is necessary to obtain a mathematically "interesting" definition. For n < 4 the range of bases is empty, so these numbers are strictly non-palindromic in a trivial way.
Properties
All strictly non-palindromic numbers beyond 6 are prime. To see why compositen > 6 cannot be strictly non-palindromic, for each such n a base b can be shown to exist where n is palindromic.
If n is even, then n is written "22" in base b = n/2 − 1.
If p = m = 3, then n = 9 is written "1001" in base b = 2.
If p = m > 3, then n is written "121" in base b = p − 1.
Otherwise p < m − 1. The case p = m − 1 cannot occur because both p and m are odd.
Then n is written "pp" in base b = m − 1.
The reader can easilyverify that in each case the base b is in the range 2 ≤ b ≤ n − 2, and the digits ai of each palindrome are in the range 0 ≤ ai < b, given that n > 6. These conditions may fail if n ≤ 6, which explains why the non-prime numbers 1, 4 and 6 are strictly non-palindromic nevertheless. Therefore, all strictly non-palindromic n > 6 are prime.
