This tag contains one of the two following problems: a) to decide when a number can be represented by a quadratic form. b) to count the number of representations of an arbitrary integer by an arbitrary but fixed quadratic form.

20 questions
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### Binary Quadratic Forms of Discriminant -3

The following is a question in my textbook: Show that any positive definite binary quadratic form of discriminant $-3$ is equivalent to $f(x, y) = x^2 + xy + y^2$. Show that a positive integer $n$ ...
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### Is $X + \frac{2}{X}$ a rational quadratic form, where $X \in \mathbb{Q} \setminus \mathbb{Z}$?

First of all, I apologize for the rather silly question. This came up while I was scouring the Internet on a mathematical terminology appropriate for a concept that I need for a paper which I am ...
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### Jacobi's four square theorem

I hope that this question is appropriate here. I am Interested in the Proof for the named theorem only using modular forms. I read on Wikipedia that the Proof actually breaks down to an Identitiy for ...
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### Find the cardinality of the following set $S_n=\{(x,y)|x^2+y^2=n, \text{where }x,y,n \in \mathbb Z\text{ and }n\geq0 \}$

We are given with $$S_n=\{(x,y)|x^2+y^2=n, \text{where }x,y,n \in \mathbb Z\text{ and }n\geq0 \}$$ Now observe that $-\sqrt n\leq x\leq\sqrt n$ and similarily, $-\sqrt n\leq y\leq\sqrt n$. This ...
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### Integers represented by $x^2 + 3 y^2$ vs. integers represented by $x^2 + x y + y^2$.

How does one show that the quadratic forms $x^2 + 3 y^2$ and $x^2 + x y + y^2$ represent the same set of integers? I think it relates to a classical result of Euler about primes of form $6k+1$. In ...
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### How many ways can a quadratic form represent a prime?

Given $a,b,c,p\in\Bbb N$ with $b^2-4ac<0$ and $p$ is a prime with $\bigg(\frac{b^2-4ac}p\bigg)=1$, how many solutions $(x,y)\in\Bbb Z^2$ are there to $$ax^2+bxy+cy^2=p?$$
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### Cubes of the Form $3x^2\pm xy+5y^2$, with $x,y$ Coprime

Are there any cubes of the form $3x^2\pm xy+5y^2$, with x, y coprime ? Partly inspired by this question. I tried various computer searches of the form $|x|\le10^a$, $|y|\le10^b$ with $a+b=6$, all of ...
Prove that $r_4(n) = 8\sum_{d|n,4\nmid d} d$ using Ramanujan's1 $\psi$1 formula. I am a little stuck here and as just wondering if I could get some advice. My work with Jacobis triple product is a ...