Strictly non-palindromic number

A strictly non-palindromic number is an integer n that is not palindromic in any positional numeral system with a base b 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 (2 ≤ b ≤ 17) is:

b	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17
19 in base b	10011	201	103	34	31	25	23	21	19	18	17	16	15	14	13	12

None of these are a palindrome, so 19 is a strictly non-palindromic number.

The sequence of strictly non-palindromic numbers (sequence A016038 in the OEIS) starts:

0, 1, 2, 3, 4, 6, 11, 19, 47, 53, 79, 103, 137, 139, 149, 163, 167, 179, 223, 263, 269, 283, 293, 311, 317, 347, 359, 367, 389, 439, 491, 563, 569, 593, 607, 659, 739, 827, 853, 877, 977, 983, 997, ...

To test whether 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 (if b > 17) as:

b	18	19	above 19
19 in base b	11	10	a single-digit number

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 composite n > 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" (a palindrome) in base b = n/2 − 1 (n/2 − 1 > 2, since n > 6).

Otherwise n is odd. Write n = p · m, where p is the smallest prime factor of n. Then clearly p ≤ m (since n is composite).

• If p = m = 3, then n = 9 is written "1001" (a palindrome) in base b = 2.
• If p = m > 3, then n is written "121" (a palindrome) in base b = p − 1 (p − 1 > 2, since p > 3).

Otherwise p < m − 1. The case p = m − 1 cannot occur because both p and m are odd.

• Then n is written "pp" (the two-digit number with each digit equal to p, a palindrome) in base b = m − 1 (since p < m − 1).

The reader can easily verify that in each case (1) the base b is in the range 2 ≤ b ≤ n − 2, and (2) the digits a_i of each palindrome are in the range 0 ≤ a_i < 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.

References

• Sequence A016038 from the On-Line Encyclopedia of Integer Sequences