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.

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- July 13th 2005, 05:08 AMphgaoX, 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. - January 4th 2006, 12:13 PMearboth
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

form1.memo1.lines.add(inttostr(x));

form1.memo2.lines.add(inttostr(y));

form1.memo3.lines.add(inttostr(z));

end;

end;

end;

(1, 18, 16)

(10, 15, 17)

(19, 12, 16)

(34, 3, 9) - January 5th 2006, 08:12 PMphgao
Wow, i'll have to do some more research to understand fully your explanation. Thanks!

- January 6th 2006, 10:55 AMearboth
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