L and P are positive integers that satisfy this equation:

(L+1)³ – L³ = P²

For example, 8³ - 7³ = 13²; 105³ - 104³ = 181², and so on.

Prove that P is always expressible as the sum of squares of two consecutive positive integers.

(For example, 13 = 2² + 3²; 181 = 9² + 10², and so on.)

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 2008-04-09 03:36:06