# strictly non-palindromic number

If for a given integer $n>0$ there is no base $1 such that each digit $d_{i}=d_{k+1-i}$ of $n$ (where $k$ is the number of significant digits of $n$ in base $b$ and $i$ is a simple iterator in the range $0), meaning that $n$ is not a palindromic number in any of these bases, then $n$ is called a strictly non-palindromic number.

Clearly $n>2$ will be palindromic for $b=n-1$, and though trivially, this is also true for $b>n$.

6 is the largest composite strictly non-palindromic number. For any other $2|n$, it is easy to find a base in which $n$ is written $22_{b}$ by simply computing $b=\frac{n}{2}-1$. For odd composites $n=mp$, where $p$ is an odd prime and $m\geq p$ we can almost always either find that for $b=p-1$, $n=b^{2}+2b+1$, or for $b=m-1$ then $n=pb+p$ and written with two instances of the digit corresponding to $p$ in that base. The one odd case of $n=9$ is quickly dismissed with $b=2$.

Title strictly non-palindromic number StrictlyNonpalindromicNumber 2013-03-22 16:25:12 2013-03-22 16:25:12 PrimeFan (13766) PrimeFan (13766) 5 PrimeFan (13766) Definition msc 11A63