# Tagged Questions

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### “Descent” on binary quadratic forms?

Let's say I have the Diophantine equation $$x^2+3n^2 = y^2+3z^2. \tag{\star}$$ where $n$ is a known integer, and we're trying to determine solutions in integers $x,y,z \ge 1$. Rewrite ($\star$) ...
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### Solving $a^2+3b^2=c^2$

I'm looking for how to solve the equation $a^2+3b^2=c^2$ where $a,b,c$ are integers and $b$ is even, I'm looking for the algorithm used to solve this kind of equations, not just the solution. Regards ...
The quadratic complete symmetric homogeneous polynomial in $n$ variables $t_1,\ldots,t_n$ is defined to be $$h_2(t_1,\ldots,t_n) := \sum_{1 \leq j \leq k \leq n} t_j t_k = \sum_{j=1}^n t_j^2 + ... 1answer 90 views ### Quadratic Diophantine Equation x^2 + axy + y^2 = z^2 I have been reading about this quadratic Diophantine equation of the form x^2 + axy + y^2 = z^2 where x, y, z are integers to be solved and a is a given integer. All integral solutions are given ... 0answers 60 views ### Finding all integral solutions of a positive definite quadratic equation Let q(x_1,\ldots,x_n) be an integral positive definite quadratic form. For d\in\mathbb{N} the equation$$q(x_1,\ldots,x_n)=d$$has a finite number of integral solutions. Is there an algorithm to ... 5answers 175 views ### Looking for proof of no solution to 4-variable quadratic diophantine equation Show there are no integers a,b,c,d such that$$\begin{cases}1=9ac+3ad+3bc-16bd \\ 1=3ad+3bc+2bd \end{cases}$$Motivation: The ideal I=(3,1+\sqrt{-17}) in R=\mathbb{Z}[\sqrt{-17}] has the ... 1answer 77 views ### Can the equation ax^2+by^2=cz^2 be solved in integers (excluding trivial solutions)? a,b,c\in\mathbb{N} and are each squarefree. Is there a general solution for this equation? I found that for this equation to be soluble in integers there are three necessary and sufficient ... 3answers 118 views ### What integers can be represented by the quadratic form 4x^2 - 3y^2 - z^2? Actually, I need to find if 4x^2 - 3y^2 - z^2 = 12 is solvable. But I somehow feel that applying theory of integer representation by quadratic forms in three variables would yield quicker results... ... 1answer 56 views ### Find all (a,b,c)\in\mathbb{Z}^3 such that b^2-4ac=-20, and -|a|<b\le|a|<|c|, or 0\le b\le|a|=|c|. Find all (a,b,c)\in\mathbb{Z}^3 such that b^2-4ac=-20, and either of the following is true: -|a|<b\le|a|<|c|, or 0\le b\le|a|=|c|. We see that if (a,b,c) is a solution, then so is ... 2answers 123 views ### How to solve an equation of the form ax^2 - by^2 + cx - dy + e =0? I am trying to find out how to solve ax^2 - by^2 + cx - dy + e = 0 to get integer solutions, failing this the rational solutions. Thanks! 1answer 72 views ### finding zeroes of a quadratic form Let a,b\in\mathbb Z be squarefree with a>0. Suppose that I know that there exist (0,0,0)\neq (x,y,z) \in \mathbb Z^3 s.t. x^2-by^2-az^2=0. Is there any known algorithm to find any such a ... 6answers 374 views ### Integral solutions of hyperboloid x^2+y^2-z^2=1 Are there integral solutions to the equation x^2+y^2-z^2=1? 2answers 930 views ### Show 15x^{2} - 7y^{2} = 9 has no integer solutions I'm trying to show the quadratic binary has no integer solution. I've used the following process to transform it into a Pell's equation of the form x^{2} - Dy^{2} = M If there is a solution, then ... 2answers 357 views ### Finding positive integer solutions to n = ax^2 +by^2 - cxy How can I find the positive integer solutions to x and y, given that n, a, b and c are all positive integers, in an equation of the form:$$n = ax^2 + by^2 - cxy.$$Specifically, I want ... 3answers 171 views ### Numbers of the form x^2+axy+by^2 This book, which needs to be returned quite soon, has a problem I don't know where to start. How do I find a 4 parameter solution to the equation x^2+axy+by^2=u^2+auv+bv^2 The title of the ... 1answer 314 views ### Find all solutions of this diophantine equation of the second degree in three variables Consider the Diophantine equation Q(x,y,z)=1, where Q(x,y,z) is the quadratic form x^2+y^2-z^2. Let S \subseteq {\mathbb Z}^3 denote the set of all solutions. It is rather easy to find several ... 2answers 188 views ### A question about integral quadratic forms Hi Would you please advise me? Consider the equation below:$$ ax^2+bxy+cy^2=n $$in which a, b, c and n are integers. We then suppose that a, b, c are constant. Is there any way to find the ... 3answers 694 views ### Existence of solutions to diophantine quadratic form Is there a general result about the existence of (non-trivial) solutions of the diophantine equation:$$Ax^2 + By^2 = Cz^2 for A,B,C known positive integers, pair-wise relatively prime? What if ...
If I have a quadratic form like $y^2 - x^2 - x = k$ none of the techniques I know work because of the nasty $x$. Note that homogenizing doesn't work because a solution of $Y^2 - X^2 - X Z = k Z^{(2)}$ ...