# How to find the “smallest” 4-couples of this problem? [closed]

I need to find a certain number of couples : $j2,j3,j5,j6$ (the number is defined in a variable), to give you an idea it is approximatively $3^4 =81$ that minimises the function :

$j2^2+j3^2+j5^2+j6^2$

But my variables follow this system of equation :

   {j2 -> ConditionalExpression[
1 + C[1] + 2 C[2] + C[3] + C[4] +
C[5], (C[1] | C[2] | C[3] | C[4] | C[5]) ∈ Integers &&
C[1] >= 0 && C[2] >= 0 && C[3] >= 0 && C[4] >= 0 && C[5] >= 0],
j3 -> ConditionalExpression[
1 + C[1] + C[2] + C[3] +
C[4], (C[1] | C[2] | C[3] | C[4] | C[5]) ∈ Integers &&
C[1] >= 0 && C[2] >= 0 && C[3] >= 0 && C[4] >= 0 && C[5] >= 0],
j5 -> ConditionalExpression[
1 + C[1] + C[2] + 2 C[3] + C[4] +
C[5], (C[1] | C[2] | C[3] | C[4] | C[5]) ∈ Integers &&
C[1] >= 0 && C[2] >= 0 && C[3] >= 0 && C[4] >= 0 && C[5] >= 0],
j6 -> ConditionalExpression[
1 + C[4], (C[1] | C[2] | C[3] | C[4] | C[5]) ∈ Integers &&
C[1] >= 0 && C[2] >= 0 && C[3] >= 0 && C[4] >= 0 && C[5] >= 0]}


Do you know how I could to it ? And the more important : do you think it is feasible in a reasonable amount of time (in less than 5 minutes) given the number of couples (81) to find and the complexity of the system ?

To be more lisible, the system of equations is the following :

$$j_1 = 1+C_1+2 C_2 + C_3 + C_4 + C_5$$ $$j_3 = 1+C_1+ C_2 + C_3 + C_4$$ $$j_5 = 1+C_1+ C_2 + 2 C_3 + C_4 + C_5$$ $$j_6 = 1+C_4$$

Where the $C_i$ are positive integers

j1 = 1 + C[1] + 2 C[2] + C[3] + C[4] + C[5];

Note that those conditions need >0 and not just >=0 to satisfy your requirement that the C[i] are positive integers.