Editing Palindromic number (section)

==Other bases==
Palindromic numbers can be considered in [[numeral system]]s other than [[decimal]]. For example, the [[Binary numeral system|binary]] palindromic numbers are those with the binary representations:
:0, 1, 11, 101, 111, 1001, 1111, 10001, 10101, 11011, 11111, 100001, ... {{OEIS|A057148}}
or in decimal:
:0, 1, 3, 5, 7, 9, 15, 17, 21, 27, 31, 33, ... {{OEIS|A006995}}
The [[Fermat prime]]s and the [[Mersenne prime]]s form a subset of the binary palindromic primes.

Any number <math>n</math> is palindromic in all bases <math>b</math> with <math>b > n</math> (trivially so, because <math>n</math> is then a single-digit number), and also in base <math>n-1</math> (because <math>n</math> is then <math>11_{n-1}</math>). Even excluding cases where the number is smaller than the base, most numbers are palindromic in more than one base. For example, <math>1221_4=151_8=77_{14}=55_{20}=33_{34}=11_{104}</math>, <math>1991_{10}=7C7_{16}</math>. A number <math>n</math> is never palindromic in base <math>b</math> if <math>n/2 \le b \le n-2</math>. Moreover, a prime number <math>p</math> is never palindromic in base <math>b</math> if <math>\sqrt{p} < b < p-1</math>.

A number that is non-palindromic in all bases ''b'' in the range 2&nbsp;≤&nbsp;''b''&nbsp;≤&nbsp;''n''&nbsp;&minus;&nbsp;2 can be called a ''strictly non-palindromic number''. For example, the number 6 is written as "110" in base 2, "20" in base 3, and "12" in base 4, none of which are palindromes. All strictly non-palindromic numbers larger than 6 are prime. Indeed, if <math>n > 6</math> is composite, then either <math>n = ab</math> for some <math>1 < a < b-1</math>, in which case ''n'' is the palindrome "aa" in base <math>b-1</math>, or else it is a perfect square <math>n = a^2</math>, in which case ''n'' is the palindrome "121" in base <math>a-1</math> (except for the special case of <math>n = 9 = 1001_2</math>).<ref>{{Cite OEIS|A016038|Strictly non-palindromic numbers}}</ref><ref>{{Cite journal|last1=Guy|first1=Richard K.|author-link=Richard Guy|date=1989|title=Conway's RATS and other reversals|journal=The American Mathematical Monthly|volume=96|number=5|pages=425–428|doi=10.2307/2325149 |jstor=2325149}}</ref>

The first few strictly non-palindromic numbers {{OEIS|id=A016038}} are:

:[[0 (number)|0]], [[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 (number)|179]], [[223 (number)|223]], [[263 (number)|263]], [[269 (number)|269]], [[283 (number)|283]], [[293 (number)|293]], 311, 317, 347, 359, 367, 389, 439, 491, 563, 569, 593, 607, 659, 739, 827, 853, 877, 977, 983, 997, ...