|
#Redirect[[Palindromic number#Other bases]]
A '''strictly non-palindromic number''' is an integer ''n'' that is not [[palindromic number|palindromic]] in any [[numeral system]] with a [[radix|base]] ''b'' in the range 2 ≤ ''b'' ≤ ''n'' − 2. For example, the number [[6 (number)|six]] is written as 110 in [[binary|base 2]], 20 in [[ternary|base 3]] and 12 in [[quaternary numeral system|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 (number)|1]], [[2 (number)|2]], [[3 (number)|3]], [[4 (number)|4]], [[6 (number)|6]], [[11 (number)|11]], [[19 (number)|19]], [[47 (number)|47]], [[53 (number)|53]], [[79 (number)|79]], [[103 (number)|103]], [[137 (number)|137]], [[139 (number)|139]], [[149 (number)|149]], [[163 (number)|163]], [[167 (number)|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 number|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''<sub>''i''</sub> of each palindrome are in the range 0 ≤ ''a''<sub>''i''</sub> < ''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 [[OEIS:A016038|A016038]] from the [[On-Line Encyclopedia of Integer Sequences]]
[[Category:Recreational mathematics]]
| 