A307019 Squares which can be expressed as the product of a number and its reversal in exactly three different ways.

6350400, 43560000, 635040000, 768398400, 4356000000, 42033200400, 55847142400, 63504000000, 64780430400, 72694944400, 76839840000, 78243278400, 234101145600, 435600000000, 4203320040000, 5086017248400, 5584714240000, 6350400000000, 6363107150400, 6478043040000, 6757504230400

Offset: 1

Author

Bernard Schott, Mar 20 2019

Keywords

• nonn
• base

Comments

1) Why do all these terms end with an even number of zeros?

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.

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))).

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.

2) When can square numbers be expressed in this way in more than three different ways?

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.

We don't know if such a solution (m,a,b) exists.

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.

There is no square less than 10^24 with 4 or more different ways. - Chai Wah Wu, Apr 12 2019

Examples

			6350400 = 2520^2 = 25200 * 252 = 14400 * 441 = 44100 * 144.
43560000 = 6600^2 = 660000 * 66 = 52800 * 825 = 82500 * 528.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..826 (n = 1..70 from David A. Corneth)
• David A. Corneth, Examples of factorizations of terms as described in Name.
• Shyam Sunder Gupta, Equal Product Of Reversible Numbers

Crossrefs

Subsequence of A083406 and A083408.

Cf. A002113, A004086, A306273, A322835.

Programs

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

Extensions

Corrected and extended by David A. Corneth, Mar 20 2019

Definition corrected and entry edited by N. J. A. Sloane, Aug 01 2019