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.

Hi Would you please advise me? Consider the equation below: $$ ax^2+bxy+cy^2=n $$ in which $a, b, c$ and $n$ are integers. We then suppose that $a, b, c$ are constant. Is there any way to find the number of answers for the equation? Actually, I have already solved this equation for many different $a, b, c$ and the number of answers. And currently I'm looking for unsolved cases. By special thanks

share|improve this question
    
@user11301 Much is known about this. Start here and see the links. Also try here –  Juan S May 24 '11 at 6:48
    
@user11301: One more here –  Juan S May 24 '11 at 6:50
    
The left part of your equation is what mathematicians call a (integral) quadratic form. There is a lot of results about the integers n they can represent or not. –  ogerard May 24 '11 at 7:54
    
I edited the title of your question to make it a tiny little bit more descriptive and a lot less urgent. I hope you're OK with that... –  t.b. May 24 '11 at 10:29
    
Try $$ x^2 + 11 y^2 = n.$$ –  Will Jagy May 25 '11 at 1:56

2 Answers 2

Assume that $\Delta = b^2 - 4 a c$ is equal to $d$, if $\frac{d-1}{4}$ is an integer, or equal to $4 d$, where $\frac{d - 2}{4}$ or $\frac{d - 3}{4}$ is an integer, and in both cases assume that $d$ is not divisible by the square of any integer. If for every odd prime divisor $p$ of $n$, you can find a non-negative integer $t$ less than $p$ such that $\frac{\Delta - t^2}{p}$ is an integer, and if $n$ is even, then $\frac{\Delta \pm 1}{8}$ is an integer, then there exist integers $x, y$ satisfying $a x^2 + b x y + c y^2 = n$. If at least one solution $x, y$ exists then the number of solutions depends on whether $\Delta < 0$, in which case there are only finitely many. If $\Delta > 0$ and at least one solution exists, then there are infinitely many. If $\Delta < 0$, then the precise number should be proportional to $\sum_{m \mid n} (\frac{\Delta}{m})$, where $(\frac{\Delta}{m})$ is the Kronecker symbol, http://en.wikipedia.org/wiki/Kronecker_symbol, and we are adding over each divisor $m$ of $n$.

share|improve this answer

The first chapter of Conway's book "The Sensual Quadratic Form" tells you, in a very elegant way, how to determine which integers are represented by a given integral binary quadratic form and how many such representations there are.

share|improve this answer

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.