I initially ran this through a spreadsheet, found the values of L being 7 and 104, but no others seemed to be likely.
I choose to factorise the equation and shortly found FrankM had related thoughts. My thought was to determine if the difference between two consecutive cubes could be a square in more than two circumstances.
My factorisation suggested that I needed 3 areas of square (L) value, 3 rectangles of (L) value and a unit square. My quick diagram was going nowhere:
 x  x     x² x²   x²   but now I have add an "x"  er, these are actually "L's", and a "1".
Back at the spreadsheet I found a new "L"; 1455.
From the spreadsheet I extracted the following:
7 8 13 (ratios)
104 105 181 14.857 13.125 13.923
1455 1456 2521 13.990 13.866 13.928
The ratios, column by column, represent the relationship between vertically adjacent values within respective columns. From these ratios it seems that the next "L" is somewhere in the vicinity of 20300. Forget the spreadsheet!!
I am satisfied that somewhere many eons down the number line there are many other "L" values.
Edited on April 9, 2008, 3:45 am

Posted by brianjn
on 20080409 03:36:06 