What are all the ordered triples of positive integers (x,y,z) with x<y<z, such that their product is four times their sum?

The following MS-DOS QBasic program finds the same results a Fernando.

I actually ran it with z finishing at 100.  I have reset it to 1000 in this listing but it takes forever; thinking about a C version. 

There is however a problem with these solutions, not the solution as such, nor the program.  What is missing is an argument as why these may/not be the only solutions.  I note that the results have the following primes as factors in one form or another: 1, 2, 3, 5 and 7.

My concern for a justifiable argument is that if I start with 1, 2 and P as my factors then 1*2*P < 4*(1+2+P) no matter how large P might be.
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The program:
(Needs some tidying but it works!)

   CLS

 ' x < y < z
 ' x is also greater that 0

   m = 1
   y = 2
   z = 3

 
 FOR a = 0 TO 4   'Provides for x to be incremented


 FOR x = a TO y - 1
 FOR y = x + 1 TO z - 1
 FOR z = y + 1 TO 1000
 

 LOCATE m, 1     'Overwrite invalid results to avoid
                 ' scrolling loss of valid results

 PRINT "a"; a; "x"; x; "y"; y; "z"; z;
      

Prod = x * y * z
sum = x + y + z

Foursum = 4 * sum


 IF Prod = Foursum THEN
  ' LOCATE m, 20
 
     PRINT "prod"; Prod; "sum"; Foursum; "x,y,z"; x; y; z
  
   m = m + 1      'Having located and printed a valid
                  'result locate for next values.
 END IF

NEXT
NEXT
NEXT
NEXT

Posted by brianjn on 2007-11-23 21:11:10