# X, Y et Z.

• Jul 13th 2005, 06:08 AM
phgao
X, Y et Z.
X, Y and Z are positive integers such that X^2 + 2Y^2 + 3Z^2 = 1417.
What is the value of X + Y + Z? Find all possible solutions.

I think this needs a program/ excel but I have no idea how to set somehting out. Help would be appreciated.
• Jan 4th 2006, 01:13 PM
earboth
I add a Delphi-procedure, which calculates the possible solutions in the form (x, y, z) if you use only positive integers.

procedure TForm1.Button2Click(Sender: TObject);
var x,y,z:integer;
begin
for x:= 1 to 50 do
for y:= 1 to 45 do
for z := 1 to 40 do
begin
if (x*x+2*y*y+3*z*z=1417) then
begin
end;
end;
end;

(1, 18, 16)
(10, 15, 17)
(19, 12, 16)
(34, 3, 9)
• Jan 5th 2006, 09:12 PM
phgao
Wow, i'll have to do some more research to understand fully your explanation. Thanks!
• Jan 6th 2006, 11:55 AM
earboth
Hi, phgao,
I did a little further investigation about your equation (it's a very interesting one!): If you use real numbers, the value for x runs from - 38 < x < 38 (because of the squares in your equation the lowest value could only be zero. So when y^2 and z^2 are zeros, the highest value for the x is approximately sqrt(1417)); -27 < y < 27 and -22 < z 22 .
You'll get an infinit number of triples which could be considered as coordinates in 3-d-graph. All points create the surface of an ellipsoid.
I've tried to attach the above mentioned graph but I'm not quite sure that you got the image because at the preview I cann't detect any image at all: tri_reell.gif