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Homestrictly non-palindromic number

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# strictly non-palindromic number

If for a given integer $n>0$ there is no base $1<b<(n-1)$ 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 $0<i<(k+1)$), 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 $b=2$.

## Mathematics Subject Classification

11A63*no label found*

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