# Homework Help: Larsen problem

1. Jan 2, 2008

### ehrenfest

[SOLVED] larsen problem

1. The problem statement, all variables and given/known data
Determine all integral solutions of $a^2+b^2+c^2=a^2 b^2[/tex]. (Hint: Analyze modulo 4.) 2. Relevant equations 3. The attempt at a solution a^2,b^2,c^2 are congruent to 0 or 1 mod 4 implies that a^2,b^2,c^2 are all congruent to 0 mod 4. This implies that a,b,c are even. $$a=2a_1, b=2b_1, c=2c_1$$ Then we have [itex]a_1^2+b_1^2+c_1^2 = 4a_1^2 b_1^2$. Now it is very clear that a_1^2,b_1^2,c_1^2 are all congruent to 0 mod 4.

Let $a_1=2a_2,b_1=2b_2,c_1=2c_2$.

If we keep doing this, we get 3 decreasing sequences of positive integers that never reach zero, which is impossible.

Therefore there are no solutions.

Is that right?

2. Jan 2, 2008

anyone?

3. Jan 2, 2008

### Dick

Is there some element of this proof that you aren't confident of? Because I don't see anything to worry about.

4. Jan 2, 2008

### ehrenfest

No. I'm just not confident in my proofs in general and the word "all" in the problem statements made me think there would be at least one.

5. Jan 2, 2008

### Dick

Well, there is a=0, b=0 and c=0. But you knew that, right?

6. Jan 2, 2008

### ehrenfest

Of course :uhh:

The reason my proof does not apply to that case is because then, for example, a,a_1,a_2,... is constant sequence, nondecreasing sequence of 0s. However, if any of a,b,c are nonzero then everything in my proof applies.