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

Sun Mar 13 17:39:57 CET 2011

Although a beautiful Robert's sequence of positive integers, most likely, is finite, it is worth to note
that the lion's share of  (b,c)-palindromes (where leading 0's are not allowed) are two-digit ones.
Further, Robert proved that there exist infinitely many 3-digit (b,c)-palindromes . In spite of the fact that trivially there exist infinitely many base b 4-digit palindromes (with not fixed b),  it seems a difficult problem to prove the corresponding fact for 4-digit (b,c)-palindromes for different b,c. I found an identity:
(2n+7)*(7n +11)^3+(-2n+8)*(7n+11)^2+(3n+8)*(7n+11)+(2n+2)=
(2n+2)*(7n +16)^3+(3n+8)*(7n+16)^2+(-2n+8)*(7n+16)+(2n+7).
Unfortunately, the coefficient -2 does not allow to obtain an immediate proof of the infinitude of
4-digit (b,c)-palindromes. For n=0,...,4, I found 5 such palindromes:
[7,8,8,2]_11=[2,8,8,7]_16=10375 (there is in Robert's list);
[9,6,11,4]_18=[4,11,6,9]_23=54634; [13,2,17,8]_32=[8,17,2,13]_37=428584;
[15,0,20,10]_39=[10,20,0,15]_44=890575.
Of course, if to permit to the second and the third digits to take negative values with absolute values,
as usual, not exceed min(b-1,c-1), then we have infinitely many such (b,c)-pseudopalindromes.

Regards,

----- Original Message -----
From: Robert Israel <israel at math.ubc.ca>
Date: Monday, March 7, 2011 3:08
Subject: [seqfan] Re: Digital question; a question about the palindromes
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

>
>
> 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