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Questions tagged [representations-by-quadratic-forms]

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.

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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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When can an odd integer $d$ be represented as $d=a^2-2b^2$ with coprime integers $a,b\ $?

I found out that in a primitive pythagorean triple $$a^2+b^2=c^2$$ the difference $d=|a-b|$ (which must be odd) can occur, if and only if we can write $$d=a^2-2b^2$$ with positive coprime integers $a,...
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Group structure on the Dirichlet binary composition of quadratic forms

It is known that the Dirichlet composition of two binary quadratic forms induces an abelian group structure on the set of positive definite quadratic forms with a given discriminant. What is that ...
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On Siegel mass formula

I am interested deeply in the following problem: Let $f$ be a (fixed) positive definite quadratic form; and let $n$ be any arbitrary natural number; then find a closed formula for number of solutions ...
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Representation of integers as quadratic forms with integer coefficients

While reading the book The sensual (quadratic) form by J.H. Conway I got curious in this question. Maybe it is trivial, but I don't know how to answer it. Let $f(x,y)=ax^2+hxy+by^2$, $g(x,y)=a'x^2+h'...
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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 ...
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Jacobi's Four square problem using Ramanujans Summation formula

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 ...
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Counting the Number of Integral Solutions to $x^2+dy^2 = n$

It is a well known result that the number of integer solutions $(x,y), x>0, y\ge 0$ to $x^2+y^2 = n$ is $\sum_{d|n}\chi(d)$, where $\chi$ is the nontrivial Dirichlet character modulo $4$ such that $...
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Difference of squares - number of representations

There exists a well-known result concerning a number of representations of $n$ as a sum of two squares. Is there anything similar for a number of representations of $n$ as a difference of two squares? ...
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Representation of prime number by binary quadratic form

For wich prime numbers $p$ there exist integers $x,y$ such that $x^2+5y^2=p$? For cases $x^2+y^2$ or $x^2+2y^2$ this condition is equivalent to discriminant of the form is quadratic residue modulo p, ...
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Number of Solutions to Diophantine Equation

$(a)$ Let $c < 2\pi$ be a positive real number. Show that there are infinitely many integers $n$ such that the equation $x^2 + y^2 + z^2 = n$ has at least $c\sqrt n$ integer solutions. $(b)$ Find ...
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Numbers representable as $x^2 + 2y^2$

I need to describe all numbers of the form $x^2 + 2y^2$. So far I've reduced the problem to primes, and showed p=2 satisfies it. I've also shown that any primes mod 5 or 7 can't be a written in this ...
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Expressing a Non Negative Integer as Sums of Two Squares

I'm writing a code in C that returns the number of times a non negative integer can be expressed as sums of perfect squares of two non negative integers. ...
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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 ...
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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 ...