# [seqfan] Re: Digital question; a question about the palindromes

Robert Israel israel at math.ubc.ca
Mon Mar 7 02:08:07 CET 2011

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On Thu, 3 Mar 2011, Robert Israel wrote:

>
>
> On Thu, 3 Mar 2011, Robert Israel wrote:
>
>> For example, two-digit (b,c)-palindromes satisfy b a_1 + a_2 = c a_2 + a_1,
>> i.e. (b - 1) a_1 = (c - 1) a_2, or a_1/a_2 = (c-1)/(b-1).  Nontrivial
>> examples will require gcd(c-1,b-1) > 1.
>> For example, 45_11 = 54_9 = 49.

Note that for any b >= 3, 21_b = 2b+1 = 12_c where c = 2b-1.  Thus every
odd integer >= 7 is a two-digit (b,c)-palindrome for some b,c.  Similarly
for other patterns.  The only positive integers <= 10000 that are not
(b,c)-palindromes (where leading 0's are not allowed) are

1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 18, 20, 24, 32, 48, 60, 72, 168, 720

Robert Israel                                israel at math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel
University of British Columbia            Vancouver, BC, Canada

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