Integer solutions of a quadratic form

Question

In some cases, it is easy to solve the question of determining the integers representable by à quadratic form. Typically, for forms of the shape $ax^2+by^2=c$, it is pretty straightforward as soon as the coefficients are positive, since there are only finitely many checks to do.

Is there a general method (at least in some cases) to solve such quadratic equations in integers?

I know there are powerful methods using Hensel lemma and Hasse principle, or analytic tools like modular forms, to answer (maybe part of) this question. Are there more Arithmetic/elementary way to approach it?

Answer 1

Equation: $[ax^2+by^2=c]$

Above has solution shown below,

$x=2qb+p(a-b)$

$y=2ap-q(a-b)$

$c=(a+b)^2(ap^2+bq^2)$

where $(p,q)$ are parameters,

for, $(a,b,p,q)=(7,5,3,2)$ we get after removing common factors,

$(x,y,c)=(13,19,2988)$

Answer 2

On the internet there are solutions for $(a,b)=(1,1)$ given by:

$x=2 (m^2+7m+1)$

$y=(11m^2+2m-4)$

$c=5(5m^2+2m+2)^2$

$(a,b,m)=(1,1,2)$

$(x,y,c)=(19,22,845)$

@ Gerry Myerson inquired about the status of 'c'. RHS of equation given by 'OP' is an integer representation. So any variables $(x,y)$ used in the LHS will add up to become an integer. Basically 'c' is not a unknown but becomes a given by virtue of $(a,b,x,y) $. If 'c' is chosen arbitrarily by 'OP' than there might not be a solution to the given equation.

Answer 3

The terms of your question are a bit too vague. As I interpret it, you start with a quadratic form $f(x,y)=ax^2+by^2$, with $a,b \in \mathbf Z$ and look for the integers $c$ which are represented by $f$, i.e. such that $f(x,y)=c$. Classical interesting examples are the Pell-Fermat equation $x^2-by^2=\pm 1$, or the Fermat equation $x^2+y^2=p$, where $p$ is a prime. Now what do you mean by an "answer to the question" ?

Concerning the Fermat equation, you noticed that, if the form f has positive coefficients, "there are only finitely many [numerical] checks to do." But what is the interest ? You can check by hand that p=2, 5, 13, 17... can be represented and, before going on experimentally forever, conjecture - as Fermat showed - that an odd $p$ is representable by the form $x^2+y^2$ iff $p\equiv 1$ mod $4$. As for the Pell-Fermat equation, you know that not only does it always have solutions, but these are generated, in a certain precise way, by "minimal"ones, so that any numerical experimentation would be silly.

To follow your trend, let us come back to Fermat, who also stated that $p$ is represented by $x^2+2y^2$ (resp. $x^2+3y^2$) iff $p\equiv 1,3$ mod $8$ (resp. $p=3$ or $p\equiv 2,3$ mod $8$). It is in trying to prove Fermat's assertions that Euler discovered the phenomenon of quadratic reciprocity and formulated conjectures which ultimately culminated in Gauss'proof of the quadratic (and some higher) reciprocity laws and their connection with genera of quadratic forms. The subsequent problem of characterising the primes $p$ of the form $x^2+ny^2, n\in \mathbf N$, was only solved much later by using such formidable tools of the 20th century as class-field theory (CFT) and the theory of complex multiplication ("explicit" CFT over imaginary quadratic fields). See e.g. the marvelous book "Primes of the form $x^2+ny^2$" by David Cox. (*)

What is my point ? Since Matiyasevich and others have given a negative answer to Hilbert's tenth problem (asking for a general algorithm to decide whether a given polynomial Diophantine equation with integer coefficients has a solution in integers), we know, roughly speaking, that any diophantine question with many enough variables and high enough degree could possibly require a sui generis solution. To your question about binary quadratic forms: Is there a general method (at least in some cases) to solve quadratic equations in integers?, I don't know a general answer, but the problem treated in Cox's book suggests a preliminary requirement: Is the special equation under study interesting enough to be worth developping a specific approach?

(*)NB. You write that there are powerful methods using Hensel lemma and Hasse principle, or analytic tools like modular forms, to answer (maybe part of) [your] question. But note that Hasse's principle is a "local-global" approach (Hensel's lemma is local) which applies only to rational quadratic forms (as opposed to diophantine). As for the special modular forms used e.g. by Cox, they belong the complex multiplication.