%I #79 Aug 01 2019 01:03:23
%S 6350400,43560000,635040000,768398400,4356000000,42033200400,
%T 55847142400,63504000000,64780430400,72694944400,76839840000,
%U 78243278400,234101145600,435600000000,4203320040000,5086017248400,5584714240000,6350400000000,6363107150400,6478043040000,6757504230400
%N Squares which can be expressed as the product of a number and its reversal in exactly three different ways.
%C 1) Why do all these terms end with an even number of zeros?
%C 1.1) Is it possible to find a term that does not end with zeros? If such a term m exists, this number must satisfy the Diophantine equation m^2 = a*rev(a) = b*rev(b) = c*rev(c). No solution (m,a,b,c) with m that does not end with zeros is known.
%C 1.2) Consider now the Diophantine equation: m^2 = a*rev(a) = b*rev(b) where a is a palindrome and b is not a palindrome. For each solution (m,a,b), we generate terms (10*m)^2 of this sequence and we get: (10*m)^2 = 100 * m^2 = (100*a)*(rev(100*a) = (100*b)*(rev(100*b)) = (100*rev(b)) * (rev(100*rev(b))).
%C Example: with a(1) = 63504 = 252^2 = 252 * 252 = 144 * 441, so (m,a,b) = (63504,252,144), we obtain the 3 following ways: 6350400 = 25200 * 252 = 14400 * 441 = 44100 * 144.
%C 2) When can square numbers be expressed in this way in more than three different ways?
%C If the Diophantine equation: m^2 = a*rev(a) = b*rev(b), with a <> b and a and b not palindromes has a solution, then it is possible to get integers equal to (10*m)^2 which can be expressed as the product of a number and its reversal in exactly four different ways.
%C We don't know if such a solution (m,a,b) exists.
%C _David A. Corneth_ has found 70 terms < 6*10^15 belonging to this sequence (see links in A083408), but no square has four solutions for m^2 = k * rev(k) until 6*10^15.
%C There is no square less than 10^24 with 4 or more different ways. - _Chai Wah Wu_, Apr 12 2019
%H Chai Wah Wu, <a href="/A307019/b307019.txt">Table of n, a(n) for n = 1..826</a> (n = 1..70 from David A. Corneth)
%H David A. Corneth, <a href="/A083408/a083408.gp.txt">Examples of factorizations of terms as described in Name.</a>
%H Shyam Sunder Gupta, <a href="http://www.shyamsundergupta.com/eporns.htm">Equal Product Of Reversible Numbers</a>
%e 6350400 = 2520^2 = 25200 * 252 = 14400 * 441 = 44100 * 144.
%e 43560000 = 6600^2 = 660000 * 66 = 52800 * 825 = 82500 * 528.
%o (PARI) is(n) = {if(!issquare(n), return(0)); my(d = divisors(n), t = 0); forstep(i = #d, #d \ 2 + 1, -1, revd = fromdigits(Vecrev(digits(d[i]))); if(revd * d[i] == n, t++; if(t > 3, return(0)); ) ); t==3 } \\ _David A. Corneth_, Mar 20 2019
%Y Subsequence of A083406 and A083408.
%Y Cf. A002113, A004086, A306273, A322835.
%K nonn,base
%O 1,1
%A _Bernard Schott_, Mar 20 2019
%E Corrected and extended by _David A. Corneth_, Mar 20 2019
%E Definition corrected and entry edited by _N. J. A. Sloane_, Aug 01 2019