Multi-Base Palindromes

A number is a palindrome if it is the same number when written forward and backward. Some base-10 palindrome can be converted into other bases and still remain a palindrome. Define \(f(x)\) as the total number of bases less than \(x\) and greater than 1 that a base-10 palindrome \(x\) can be converted into and remain a palindrome.

Find the number \(n\) (in base 10) that satisfies the following conditions:

  • \(f(x)\) is maximized over the domain \(0<x<100000\) when \(x=n\in \Bbb{Z}.\)
  • \(1<n<100000.\)
  • \(n\) is a palindrome.

Details and Assumptions:

  • \(x_a\) means that \(x\) is written in base \(a;\) for example, \(101_2=5_{10}\).
  • \(33_{10}=1000001_2\) and \(33_{10}=11_{32}\).
    So 33 is a palindrome in base 2, 10, and 32, and thus \(f(33)=3\).
  • \(1,6,8,0_{10}=40,40_{41},\) which means \(1680\) is a palindrome in base 41 because on the RHS, each 40 represents a single digit. \((\)This example uses a notation that separates digits of a number using commas. For example, \(5,5,5_{10}=555_{10}.)\)
  • Do not pad the start of the number with 0's. For example, 10 is not a palindrome, though it could be written as 010.

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