Strictly non-palindromic number

Encyclopedia : S : ST : STR : Strictly non-palindromic number

A strictly non-palindromic number is an integer n that is not palindromic in any numeral system with a base b in the range 2 ≤ b ≤ n − 2. For example, the number six 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.

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

1, 2, 3, 4, 6, 11, 19, 47, 53, 79, 103, 137, 139, 149, 163, 167, 179, 223, 263, 269, 283, 293, …

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.

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 must 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.

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

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.

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.

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 from the On-Line Encyclopedia of Integer Sequences

From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.