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

[Vladimir Shevelev]

I give a general sufficient condition of the two-digit palindromeness of n. Let f,g,h  are positive integers with the condition g<h<=f*g, and let d=f*g*h+g+h  is a  divisor of n not less than n*h/f*g.
Then n is two-digit palindrome with b=f*h+1, c=f*g+1, such that n=[n*g/d, n*h/d]_b=
[n*h/d, n*g/d]_c.
Indeed, since g<h, we should show that digits do not exceed f*g. By the condition, we have n*g/d<=n*g*f*g/n*h=f*g^2/h<=f*g*h/h=f*g and n*h/d<=n*h*f*g/n*h=f*g. Now the palindromeness of n follows from the identity g*(f*h+1)+h=h*(f*g+1)+g (the multiplication by n/d gives both of representations of n). In particular, if n=d=f*g*h+g+h, then n=[h,g]_(f*g+1)=[g,h]_(f*h+1) and to show that every odd number>5 is a (b,c)-palindrome, we can take f=b-2, g=1,h=2. I think even that, CONVERSELY, if n is a two-digit palindrome, then it has the above representations n=[n*g/d, n*h/d]_b=[n*h/d, n*g/d]_c  for some values of f,g,h with conditions:  g<h<=f*g, and f*g*h+g+h  is a  divisor of n not less than n*h/f*g.
 
Thanks, Robert, for sequence of ALL non-(b,c)-palindromes up to 10^4. I think that it should be published in OEIS. It is interesting that, by its form, it is a little similar to the Euler sequence
of 65 Numeri Idonei (Euler continued his calculations up to 10^4 and did not find other terms). As well known now, the 66-th term>1848 does not exist. And what is about your sequence?
 
Regards,
Vladimir


----- 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 
> Columbia            Vancouver, BC, Canada
> 
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> 

 Shevelev Vladimir‎