strictly non-palindromic number

If for a given integer n>0 there is no base 1<b<(n-1) such that each digit di=dk+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 numberMathworldPlanetmath 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 22b by simply computing b=n2-1. For odd composites n=mp, where p is an odd prime and mp we can almost always either find that for b=p-1, n=b2+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
Canonical name StrictlyNonpalindromicNumber
Date of creation 2013-03-22 16:25:12
Last modified on 2013-03-22 16:25:12
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 5
Author PrimeFan (13766)
Entry type Definition
Classification msc 11A63