No Integral Solutions to Larsen Problem

[ehrenfest]

[SOLVED] larsen problem

Homework Statement

Determine all integral solutions of [itex]a^2+b^2+c^2=a^2 b^2[/tex]. (Hint: Analyze modulo 4.)

Homework Equations

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 $a_1^2+b_1^2+c_1^2 = 4a_1^2 b_1^2$. 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?

 

[ehrenfest]

anyone?

 

[Dick]

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

 

[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.

 

[Dick]

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

 

[ehrenfest]

Of course

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.