Homework Help: Algebraic Number Theory Question

1. Nov 9, 2011

putongren

This is actually a Number Theory question, but requires expertise that doesn't go beyond simple algebra.

1. The problem statement, all variables and given/known data
Show that (1+xy)(1+zy)(1+zx) is a perfect square iff (1 + xy), (1+yz) , and (1+zx) are perfect squares.

2. Relevant equations

3. The attempt at a solution

I initially tried to solve it like this

let 1 + xy = s2 , 1 + yz = t2, 1 + zx = u2. I substituted the new variables and got

(1+xy)(1+zy)(1+zx) = s2t2u2 - s2t2+t4-t2+s2

I tried to get the RHS of the equation to be in some square form, but couldn't. I thought that introducing new variables to the problem would add to the complexity of the problem, but I thought that adding new variables would give me a new way to solve the problem also.

So .. any thoughts?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Nov 9, 2011

Hurkyl

Staff Emeritus
What are x,y,z,s,t,u? Are you working on the 'if' part or the 'only if' part? (or have you not gotten that far yet?) Where did that equation come from?

3. Nov 9, 2011

putongren

forgot to mention x,y,z are positive integers,

I just defined s,t,u as the number whose square is xy + 1, yz +1, zx + 1 respectively

4. Nov 9, 2011

SammyS

Staff Emeritus
That last equation is obviously wrong.

(1+xy)(1+zy)(1+zx) = s2t2u2 , which is a perfect square.

That's the easier half of the proof.

Now show that if (1+xy)(1+zy)(1+zx) = w2, then each of (1+xy), (1+zy), and (1+zx) are perfect squares.

5. Nov 9, 2011

putongren

Yeah, I suspect it was wrong, but I still have no idea how to show that (1 + xy)(1 + zy)(1 +wz) = w2.

6. Nov 9, 2011

SammyS

Staff Emeritus
Did you try multiplying the left side out? -- Then try refactoring it.

Last edited: Nov 9, 2011