# Tagged Questions

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### A conjecture on products/composition of Pell forms

Based on a few brute-force calculations, I've formulated the following. Conjecture. Let $x,y,u,v,p,q,a,b,c \ge 2$ be integers such that $$(x^2+ay^2)(u^2+bv^2) = p^2+cq^2,$$ and write \begin{align} ...
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### Divisors of numbers of the form $a^2+2b^2$ with $\gcd(a,b)=1$

Let's say I have a number $n$ which can be written as $a^2+2b^2$ for integers $a,b$. By Fermat/Euler/etc., I know that the primes dividing the squarefree kernel of $n$ cannot be congruent to $5$ or ...
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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 ...
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### Primes of the form $x^2+ny^2$ where $n\equiv 1\pmod{4}$ is a squarefree number

Let $n\equiv1\pmod{4}$ be a squarefree number and $p\equiv1\pmod{4n}$ be a prime number. Does there exist $x,y\in\mathbb{N}$ such that $p=x^2+ny^2$?
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### A characterization of an ambiguous class of binary quadratic forms of discriminant $D$

We use the definitions of this question. Let $D$ be a non-square integer such that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). There exists a bijection $\psi\colon Cl^+(R) \rightarrow C(D)$ by ...
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### 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 ...
### Representing a number as a sum of at most $k$ squares
Fix an integer $k >0$ and would like to know the maximum number of different ways that a number $n$ can be expressed as a sum of $k$ squares, i.e. the number of integer solutions to  n = x_1^2 + ...