Questions tagged [binary-quadratic-forms]
A binary quadratic form is a quadratic form in two variables.
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Reference request: analogue of Cramér's conjecture for integers represented by binary quadratic form
The Prime Number Theorem asserts that $\pi(N) \sim N/\log N$ as $N \to \infty$ where $\pi(N)$ is the prime counting function. Colloquially, the (average) density of primes $\le N$ is like $1/\log N$. ...
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On $a_n(x)=\sum_{i,j=0}^n \binom ni^2\binom nj^2\binom{i+j}ix^{i+j}$ (III)
As in Question 491655 and Question 491762, we define
$$a_n(x):=\sum_{i,j=0}^n\binom ni^2\binom nj^2\binom{i+j}ix^{i+j}$$
for each nonnegative integer $n$.
Here we pose some curious congruences ...
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On $a_n(x)=\sum_{i,j=0}^n\binom ni^2\binom nj^2\binom{i+j}ix^{i+j}$ (I)
Recall that the Apéry numbers are given by
$$A_n=\sum_{k=0}^n\binom nk^2\binom {n+k}k^2\ \ (n\in\mathbb N=\{0,1,2,\ldots\}).$$
In a 2012 JNT paper I conjectured that for any odd prime $p$ we have
$$\...
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Why is the coefficient of $\sqrt{D}$ in the fundamental unit relatively smooth?
Let $D$ be a non-square discriminant, in the sense that $D \equiv 0,1 \pmod{4}$ and $D$ is not itself a square. There has been much work on the fundamental unit $\epsilon_D$, by which I mean the ...
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Sum of representation functions of binary quadratic forms
Given a positive definite binary quadratic form $f(x,y)=ax^2+bxy+cy^2$ of fundamental discriminant $-D$ with $D>0$, let $$r_f(n)=\#\{(x,y)\in\mathbb{Z}^2 | f(x,y)=n\}.$$
Let's say I have another ...
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Quadratic forms with the same roots over GF(2) for low rank problems
Let $Q_1(x)=x^TA_1x$ and $Q_2(x)=x^TA_2x$ with $x\in GF(2)^n$, $A_i\in GF(2)^{n\times n}, i \in \{1, 2\}$. If $rank(A_1)=rank(A_2)=2$, is it possible that $Q_1(x)$ and $Q_2(x)$ can have the same roots ...
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Representing positive integers $n$ by binary forms $n=ax^2+by^2$, $a\geq 0$, $b\geq 0$
In there a finite set of $(a_k,b_k)\in\mathbb{Q}_{\geq 0}\times \mathbb{Q}_{\geq 0}$ so that any $n\in \mathbb{Z}_{>0}$
can be written as $n=a_k x^2+b_k y^2$ for some $k$ ?
This is related to ...
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Genus of binary quadratic forms: $f(x,y), g(x,y)$ in same genus if and only if represent same values in $(\mathbb Z/m\mathbb Z)^\ast$ for all $m$
In David Cox's book: Primes of the form $x^2+ny^2$, second edition, there is a theorem(Theorem 3.21, page 52) characterize whether two binary quadratic forms in the same genus. The contents of the ...
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On the parity of the sum $\sum_{1\le j<k\le p-1\atop p\nmid aj^2+bjk+ck^2}(aj^2+bjk+ck^2)$
QUESTION. Let $p$ be an odd prime and let $a,b,c\in\mathbb Z$. How to determine the parity of the sum
$$S_p(a,b,c)=\sum_{1\le j<k\le p-1\atop p\nmid aj^2+bjk+ck^2}(aj^2+bjk+ck^2)$$
in terms of $a,b,...
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On an integer factoring algorithm based on smooth class number of quadratic fields
We got an algorithm and toy implementation of integer factoring algorithm
based on smooth class number of quadratic fields.
It is close to the elliptic curve factorization method (ECM) and
succeeds if ...
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On a theorem by Iwaniec about binary quadratic polynomials representing infinitely many primes
In Theorem 1.1 (i) of http://matwbn.icm.edu.pl/ksiazki/aa/aa24/aa2451.pdf, Iwaniec showed that a certain type of quadratic polynomial $P(x,y)$ represents infinitely many primes, where $(x,y) \in \...
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Could efficient solutions of $x^2+n y^2=A$ be related to integer factorization?
Let $n$ be positive integer with unknown factorization and $A$ integer with known
factorization.
According to pari/gp developers pari can efficiently find all solutions of:
$$x^2+n y^2=A \qquad (1)$$
...
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heights of ideal classes and reduction theory for Bhargava cubes
Suppose $K$ is a quadratic imaginary field with discriminant $D$; let $S$ denote the ring of integers in $K$. For a fractional $S$-ideal $J$, define the height of $J$, denoted $H(J)$, to be the ...
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A cubic equation, and integers of the form $a^2+32b^2$
I am trying to determine whether there are any integers $x,y,z$ such that
$$
1+2 x+x^2 y+4 y^2+2 z^2 = 0. \quad\quad\quad (1)
$$
It is clear that $x$ is odd. We can consider this equation as quadratic ...
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Are there natural isomorphisms $S^{(2,1)}(k^{m+1})\cong k^2\otimes W$?
In this popular 2019 MO question, user მამუკა ჯიბლაძე asked:
The spaces $\operatorname{S}^2(k^n)$ and $\Lambda^2(k^{n+1})$ from the title have equal dimensions. Is there a natural isomorphism between ...
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