Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Is there a general result about the existence of (non-trivial) solutions of the diophantine equation:

$$Ax^2 + By^2 = Cz^2$$

for A,B,C known positive integers, pair-wise relatively prime?

What if we know C is 1 or 3?

share|improve this question
1  
this is the same as asking for rational points on an ellipse. you can get a rational parameterization of the ellipse by fixing a point on the ellipse and finding the other point of intersection of a line through the given point. so for rational slopes through one of the obvious (trivial) points, you get infinitely many solutions. –  yoyo Mar 7 '11 at 22:38
    
But then I'd need to find a point on the elipse, and I'm looking for whether there are rational points on the ellipse. Once I've found one such point, I know that I can find infinitely many, yes :) –  Thomas Andrews Mar 7 '11 at 22:42
1  
For example, if (A,B,C)=(5,2,1), there is no solution. –  Thomas Andrews Mar 7 '11 at 22:48
add comment

3 Answers

up vote 6 down vote accepted

Yes; actually, this is one of the only classes of Diophantine equations for which such a result exists! First we will make some simplifying observations. Observe that it is pretty easy to tell what happens when $z = 0$, so suppose $z \neq 0$. Next observe that finding integer solutions is equivalent to finding rational solutions, and since we can scale all three variables by the same constant we can assume $z = 1$, so we are solving $Ax^2 + By^2 = C$ for rationals $x, y$.

It's a classical result that if there is one solution, there is a straightforward way to describe all of the other solutions: if $(x_0, y_0)$ is a solution, then any line of the form $(x_0 + at, y_0 + bt)$ where $a, b$ are fixed rationals intersects the curve $Ax^2 + By^2 = C$ in exactly one other point, and this intersection must be rational; conversely, every other rational solution arises in this way.

So it suffices to find a single solution. To do this the key is the Hasse-Minkowski theorem, which tells you that solutions exist over $\mathbb{Q}$ if and only if they exist over $\mathbb{R}$ and over the p-adic numbers $\mathbb{Q}_p$ for all primes $p$.

It is very easy to check if a solution exists over $\mathbb{R}$, so it suffices to check if solutions exist over $\mathbb{Q}_p$ for all $p$. If $p \nmid 2ABC$, then the Chevalley-Warning theorem shows that the equation has a solution in $\mathbb{Z}/p\mathbb{Z}$, and by Hensel's lemma these solutions can be upgraded to solutions in $\mathbb{Z}_p \subset \mathbb{Q}_p$.

So we are reduced to checking the finitely many primes dividing $2ABC$. But for any particular such prime, this is more or less an application of quadratic reciprocity together with Hensel's lemma again.

This is classical material; I think you can find a more thorough exposition in the beginning of Cassels' Lectures on Elliptic Curves.

share|improve this answer
    
Thanks. My mathematics is rusty, but I had a feeling this was possible. –  Thomas Andrews Mar 7 '11 at 22:53
1  
Let me take the opportunity to mention something which I think is amazing: there is no such result for polynomials of degree $3$ or higher. We can't even prove such a result for the specific case of polynomials of degree $3$. There is a nice discussion of what is known (although it may be outdated by now) in Poonen's Computing Rational Points on Curves: citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.10.2007 –  Qiaochu Yuan Mar 7 '11 at 23:25
    
It seems like the final hard part is the case p=2. You cannot use Hensel's lemma mod 2, can you? –  Thomas Andrews Mar 8 '11 at 19:16
    
@Thomas: there is a more general version of Hensel's lemma which works here (and this is the version stated e.g. in the Wikipedia article). You need to verify if there are solutions mod 8, I think. –  Qiaochu Yuan Mar 8 '11 at 19:20
add comment

As Qiaochu mentioned, this is a classical problem that is a prototypical example of an equation that is amenable to the local-global approach. Legendre obtained a complete solution by descent circa 1785. Later, as $p$-adic methods emerged, it was realized that Legendre's solution could be elegantly reformulated via such local-global techniques. You can find a nice readable four-page introduction on pp. 238-242 of Harvey Cohn: Advanced Number Theory.

share|improve this answer
add comment

I would happily add Serre's book.

share|improve this answer
    
It would probably be a good idea to specify which book; I guess you mean A Course in Arithmetic. –  Qiaochu Yuan Mar 7 '11 at 23:18
    
yes. It is that book. –  Kerry Mar 8 '11 at 15:01
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.