Find all natural numbers which can be written in the form (a+b+c)²/(abc) where a,b,c are also natural numbers.

(In reply to re(2): Some were discovered.... by Charlie)

To extend Charlie's finds using the patterns I discovered 26 additional sets were found...
   2     578    3364     4
   2     726    2704     3
   3     507    3468     3
   3    1089    4056     2
   4     242    3362     4
   4    1156    6728     2
   5    1620    4225     1
   6     242    3844     3
   6     588    2178     1
   6    2178    8112     1
   8     400    2312     1
   8     484    6724     2
   8    2312   13456     1
   9     363    5766     2
   9    1521   10404     1
  12      75    2523     3
  16      54    2450     3
  16     968   13448     1
  18     150    2352     1
  18     726   11532     1
  20     405    7225     1
  24      81    3675     2
  25     405    9245     1
  36     225    7569     1
  45     100    4205     1
  48     162    7350     1

It can be noted that a can exceed 25. (By extension, values of 36, 45 and 48 for a were found.) Of course a, b and c can be interchanged, but for the purpose of finding the natural numbers of (a+b+c)2/(abc), we define c >= b >= a >=1. As Brian Smith stated, there may be no limit to what values a, b and c can take to yield an integer. Following is another 3 finds where, in one, c has a value of 71289....
   1    1156    7921     9
   3    3468   23763     3
   9   10404   71289     1

Edited on October 16, 2007, 7:37 am

Posted by Dej Mar on 2007-10-16 00:09:28