Strictly non-palindromic number

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 OEIS|id=A016038 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"&thinsp;&middot;&thinsp;"m", where "p" is the smallest odd prime factor of "n". Then clearly "p" &le; "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 &le; "b" &le; "n" − 2, and (2) the digits "a""i" of each palindrome are in the range 0 &le; "a""i" < "b", given that "n" > 6. These conditions may fail if "n" &le; 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


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