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When palindromes are numbers

I suppose we all know what a palindrome is; it is a bit of text that reads the same (ignoring spaces) backwards as forwards. For example, "Madam, I'm Adam" and "A man, a plan, a canal, Panama". We don't always think of numbers as bits of text, but they are, and there are numbers that are palindromes, for example, 101 and 121.

When we talk about numbers, however, we have to say what base we are using. A number can be a palindrome in one base but not in another. For example 101 base 10 is 65 base 16. This suggests a little puzzle:

Are there any numbers that are not palindromes in any base, with the proviso that the palindrome must have at least two "digits"? If so, which is the smallest? Is there a largest, and, if so, what is it?

Solution


This page was last updated Mar 4, 2011.

Richard Harter's World
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2011
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