Can You Order Quadratic Sums with Natural Numbers and Rational Coefficients?

[Eratosthenes]

Hi I have a question I will try to explain it as best as I possibly can.

Ok I have sum that look like this:

(a*x^2 + b*y^2 + c*z^2)

a, b, c are constants and are positive rational numbers
x, y, and z are natural numbers

I am trying to find a method to choose values for x, y, and z so I can find all the sums in increasing order. a, b, and c are fixed constants.

So for example if a = 1, b = 1/4, c = 1/9,

then I have: (1*x^2 + (1/4)*y^2 + (1/9)*z^2)


1st Sum: 1.36, with x = 1, y = 1, z = 1
2nd Sum: 1.694 with x = 1, y = 1, z = 2
3rd Sum: 2.11 with x = 1, y = 2, z = 1
4th Sum: 2.25 with x = 1, y = 1, z = 3
5th Sum: 2.44 with x = 1, y = 2, z = 2
6th Sum: 3 with x = 1, y = 2, z = 3
7th Sum: 3.028 with x = 1, y = 1, z = 4
8th Sum: 3.36 with x = 1, y = 3, z = 1
9th Sum: 3.472 with x = 1, y = 3, z = 2
10th Sum: 3.778 with x = 1, y = 2, z = 4


Of course I can different values for a, b, and c. Let's say a = 1/4, b = 2, c = 1/16 so then I'd have:

((1/4) * x^2 + 2*y^2 + (1/16)*c^2), and I'd have to find values for x, y, and z so that the sums would be in increasing order.


Is it possible? And if so any ideas, suggestions? I figure maybe someone here may be familiar with these types of problems. This isn't from a math book or from a math course so I'm not sure where to start. It came up in something else I was doing and I figured it would be fun to find a solution, I just don't want to run around in circles forever if it isn't possible heh. I don't really have a starting point. I tried to find a pattern but I didn't see one. I hope I didn't make any mistakes, it's really late. Thanks.


Edit: If I put this in the wrong section feel free to move it. I am not sure where this belongs so I just posted in general.

 

[TenaliRaman]

hmm..
well since a,b and c are all non-negative and x,y,z are also non-negative
Therefore set y,z to zero and keep incrementing x. The sum will always be increasing.
(Infact fix any two of (x,y,z) to any non-negative value and keep on incrementing the third variable, as long as the coefficient of this third variable isn't zero).

-- AI
P.S -> Am i missing something?

 

[Eratosthenes]

hmm..
well since a,b and c are all non-negative and x,y,z are also non-negative
Therefore set y,z to zero and keep incrementing x. The sum will always be increasing.
(Infact fix any two of (x,y,z) to any non-negative value and keep on incrementing the third variable, as long as the coefficient of this third variable isn't zero).

-- AI
P.S -> Am i missing something?

Yes my fault terribly sorry. a, b, c are positive rational numbers, and x, y, z are positive integers, so yea nothing can be zero in my problem. I wasn't given this problem I created it and I am using it to solve another larger problem so sorry for the mistakes. Thank you for reading and trying to help.

 

[bao_ho]

output, then sort.

a=1 : b=1/4 : c=1/9

for z=1 to Zmax step 1
for y=1 to Ymax step 1
for x=1 to Xmax step 1
Sum = (a*x*x + b*y*y + c*z*z)
append table, Sum, x, y, z
next x
next x
next x

Sort table, Sum

 

[bao_ho]

a=1 : b=1/4 : c=1/9

for z=1 to Zmax step 1
for y=1 to Ymax step 1
for x=1 to Xmax step 1
Sum = (a*x*x + b*y*y + c*z*z)
append table, Sum, x, y, z
next x
next y
next z

Sort table, Sum