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Given

$$(s-x_{1})(s-y_{1})(s-z_{1}) = (s-x_{2})(s-y_{2})(s-z_{2})$$

Where $s=x_1+y_1+z_1$ and all variables are positive non-zero integers.

I need to prove that such values of $x_{1}, x_{2}, y_{1}, y_{2}, z_{1}, z_{2}$ can or can't exist.

I have tried to perform an expansion of both sides but this produced a complex expression.

Please use simple terms to describe the solution.

Note: This is not a homework. I am too old for those now...

Thank you for your help.

Problem Origins

A question was asked where there was a need to create a database which rows could be stored but only when there are no permutations in the database for 3 given columns. That is, if a row such as $3,5,1$ exists you can't store $1,5,3$.

I thought this can be solved if I could find a function $f(x,y,z) = v$ such that when you use any of the 6 permutations of $x,y,z$, the function result is always the same. That is, $v=f(x,y,z)=f(x,z,y)=f(y,x,z)=...$

I though since I have 3 values, I may use the area of a triangle using the points $(x,p), (y,p+1)$ and $(z,p+2)$ where $p$ is an arbitrary integer. The area can be easily calculated this way and it satisfies the above requirement. However, another requirement was to prove that the function does not produce the same value $v$ for any set of numbers other than $x,y,z$ or their permutations. We know that we can have 2 triangles of the same area and different side lengths. So I tried to add that restriction in and got into above question. If we can prove that the above equality does not hold, the the area of a triangle can be used to provide the desired function.

EDIT-1 Thank you for your all your help. As @André Nicolas mentioned, the constraints are:

$s1=x_1+y_1+z_1 = s2=x_2+y_2+z_2$

Also, the points $(x_1,y_1,z_1)$, $(x_2,y_2,z_2)$ can't be permutations of one another. For example the points (1,1,2) and (1,2,1) don't represent a solution.

Please accept my apologies for missing this in the original version.

Edit-2 - Answer

I used brute force and found that indeed some points satisfy the requirement of an equality.

For example:

(1,5,5) and (2,3,6)

for (1,5,5): s1=11, lhs product=10*6*6=360

for (2,3,6): s2=11, rhs product=9*8*5=360

and the 2 points are not permutations of the same set of numbers.

Thanks all for your help.

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2  
If your entries are integers $x,y,z$ you could use the function $f(x,y,z) = x+y+z + \sqrt{2}(xy + yz + xz) + \sqrt{3} (xyz)$ –  Cocopuffs Aug 14 '12 at 11:05
    
@EmmadKareem: So you want $x_1+y_1+z_1=x_2+y_2+z_2$ (call this sum $s$ like you did) and the products equal as in your post? Or do you merely want $x_1+y_1+z_1=s$ and products equal, with no restriction on the sum $x_2+y_2+z_2$? –  André Nicolas Aug 14 '12 at 15:34
    
@Cocopuffs, this is clever, could you please tell me about how you came across this formula? Also, you may want to add this as an answer. –  Emmad Kareem Aug 14 '12 at 21:37

3 Answers 3

up vote 2 down vote accepted

Here's what I meant in my comment.

Assume $x,y,z \in \mathbb{Q}$. The tuple $(x,y,z)$ is determined uniquely up to order by the polynomial $$(X - x)(X - y)(X - z) = X^3 - (x + y + z)X^2 + (xy + xz + yz)X - (xyz).$$ Since $1, \sqrt{2}, \sqrt{3}$ are linearly independent in $\mathbb{R}$ over $\mathbb{Q}$, the function $$f(x,y,z) = (x+y+z) + \sqrt{2} (xy + xz + yz) + \sqrt{3}(xyz)$$ can never take the same value for different values of $x+y+z, xy+xz+yz, xyz \in \mathbb{Q}$. So $f(x_1,y_1,z_1) = f(x_2,y_2,z_2)$ if and only if $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ are permutations of each other.

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Excellent and clever. Thanks. –  Emmad Kareem Aug 15 '12 at 9:54
    
I guess your formula above could be used to tell if two sets $(x_1,y_1,z_1) and (x_2,y_2,z_2)$ are permutations or not. You may want to generalize it. –  Emmad Kareem Aug 16 '12 at 4:11
    
@EmmadKareem What do you mean? –  Cocopuffs Aug 16 '12 at 5:47
    
I mean using your formula one could verify that given 2 set of numbers like (1,3,2) and (3,2,1), one can tell that the 2 sets represent a permutation of the numbers 1,2,3. From a software view point this is interesting because the alternative would be to write an algorithm for doing that. However, using your formula, it is easy to compare the result by evaluating the function for each set. If the value is the same, then they are permutations of the base set. –  Emmad Kareem Aug 16 '12 at 6:15

Regarding the original problem: Can't you just arrange the 3 values in ascending order? In that way f(x,y,z) = f(x,z,y) = ...etc always. And it's injective, 1-1, as you needed it.

I'm afraid I have no idea about the equation though.

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Could you please clarify? Thanks. –  Emmad Kareem Aug 14 '12 at 21:36

Surely you take any number that can be represented as the product of $3$ discrete numbers in $2$ different ways, and set $s$ as the sum of one set of products.

For example:

$48 = (1 \times 4 \times 12) = (2 \times 3 \times 8)$.
Let $s$ = $(2+3+8)=13$.
$(13-12)(13-9)(13-1) = (13-11)(13-10)(13-5)$

Or have I misunderstood something?

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Thank you for your help, please see the edit to the original post. –  Emmad Kareem Aug 14 '12 at 21:35

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