# 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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