20
votes
Accepted
Is the diophantine equation $3x^2+1=py^2$ always solvable for each prime $p\equiv 13\pmod{24}$?
Updated on 2019/08/21: I prove Conjectures 1-3 below. I will use the usual terminology and notations for binary quadratic forms. In particular, I will use the first two pages of Pall: Discriminantal ...
- 85.6k
20
votes
Accepted
reference for: no finite set of positive (integer) binary quadratic forms represents all primes
This is indeed correct; I don't know a reference, but here's a proof. Let ${\mathcal D}$ be a finite set of $K$ negative fundamental discriminants. We want to show that the set of primes not ...
- 42.4k
19
votes
Accepted
Upper bound on answer for Pell equation
Let $d$ be a positive fundamental discriminant, $\epsilon_d$ denote the fundamental unit, $h(d)$ the class number, and $\chi_d$ the primitive character associated to the discriminant $d$. The class ...
- 42.4k
18
votes
Accepted
Infinite cyclic subgroups of $\text{SL}_2(\mathbb{Z})$
The maximal infinite cyclic subgroups are of the form you mentioned (to get all infinite cyclic subgroups you also need to include their finite index subgroups). This follows from Theorem 1.4 in ...
- 3,860
16
votes
Accepted
Primes of the form $x^2 + y^2 + 1$
Iwaniec (Acta Arith. 1972) showed that the number of such primes has the order $x/(\log x)^{3/2}$. His result applies more generally to translates of binary quadratic forms. One can also show that ...
- 42.4k
15
votes
$x^2+7y^2=2^n$ and sums of four squares
The Diophantine equation $x^2 + 7y^2 = 2^n$ is "trivial" in the sense that its solutions can be described in a very simple way (see boxed formula near the end for solutions in odd numbers ...
- 41.8k
12
votes
Upper bound on answer for Pell equation
Let $u_0=x_0+y_0\sqrt{p}$ be the smallest solution with $x_0,y_0>0$. (I assume you're talking about an upper bound for the smallest solution, since obviously there are solutions that are ...
- 42.3k
11
votes
Accepted
Equidistribution of CM points in the principal genus
This is known, and follows from Theorem 2 in Harcos and Michel's paper The subconvexity problem for Rankin-Selberg $L$-functions and equidistribution of Heegner points. II (Invent. math., vol. 163, ...
- 13.5k
10
votes
Equidistribution of CM points in the principal genus
Vesselin Dimitrov gave a nice answer, but let me point out that for the OP's question one does not need the result of Harcos-Michel (2006). Instead, the original work of Duke (1988) or alternatively ...
- 85.6k
10
votes
Accepted
The mean value of $y \log{y}$ over the ordinates of the CM points
Let $g : \Gamma \backslash \mathbb{H} \to \mathbb{C}$ be any bounded continuous function. Duke's theorem states that
\[\frac{1}{h(D)} \sum_{A \in \mathrm{Cl}_K} g(z_A) = \frac{1}{\mathrm{vol}(\Gamma \...
- 7,459
10
votes
Accepted
A conjecture for primes $p\equiv\pm1\pmod5$
Let us consider $\mathbb{Q}(\sqrt{5})$ as a subfield of $\mathbb{R}$. Let us also consider the positive fundamental unit $\epsilon:=(1+\sqrt{5})/2$, whose square generates the group of totally ...
- 85.6k
9
votes
Equivalence of binary quadratic forms over $\operatorname{GL}_2(\mathbb{Q}_p)$ or $\operatorname{GL}_2(\mathbb{Z}_p)$
The number of classes of (regular) binary quadratic forms over $\mathbb{Q}_p$ equals $7$ when $p\neq 2$, and it equals $15$ if $p=2$. See Section IV.2.3 of Serre: A course in arithmetic.
The number ...
- 85.6k
8
votes
Accepted
Representation of two related integers by the same binary quadratic form
This question can be answered by general theory, at least when $\Delta$ is a fundamental discriminant (so it's not a square times a smaller discriminant). Assuming this, the general procedure for ...
- 18.7k
8
votes
Accepted
Representation of integers by principal binary quadratic forms
There are precisely two such proper divisors of $k$, and their product equals $-k$. This result is essentially due to Gauss, see Theorem 1 in Pall: Discriminantal divisors of binary quadratic forms, J....
- 85.6k
8
votes
Accepted
On binary quadratic forms which are not proper subforms of another binary quadratic form
Yes. It is about equivalence over $SL_2 \mathbb Z.$ Your form $f$ (primitively) represents some value not divisible by the fixed prime $p.$ Indeed, from original coefficients $\langle a,b,c \rangle$ ...
- 24.5k
7
votes
The mean value of $y \log{y}$ over the ordinates of the CM points
Can we do this by reducing it to a sum over ideals?
By Duke's theorem, we can easily deal with the low $y$ terms. Thus we can adjust the boundary condition to $-a < b \leq a < \sqrt{- D /3 }$.
...
- 116k
5
votes
Accepted
A class number estimate
This is from Buell, Binary Quadratic Forms, pages 109-119.
Note that $h(-16) = h(-4) = 1.$
When $p$ is an odd prime, we use the Legndre symbol in
$$ h(-4p^2) = \frac{p - (-1|p)}{2} $$
After that, ...
- 24.5k
5
votes
Accepted
Recursions for some binary theta series in characteristic 3
We establish the recursion for all $n$ by writing
the rank-2 theta series $A(n)$ in terms of the rank-1 thetas
$$
S(q) := \sum_{m \in \bf Z} q^{m^2} = 1 + 2q + 2q^4 + 2q^9 + \cdots,
$$ $$
T(q) := \...
- 71.6k
4
votes
Accepted
Ray class groups through binary quadratic forms
I guess that the answer is no. The binary forms with discriminant
$\Delta \cdot f^2$ describe ring class fields modulo $f$ (see Cox's book), so by taking the limit as $f \to \infty$ something like ...
- 30.6k
4
votes
On the equivalence of a pair of binary quadratic forms
The exact class number formula was given by Jorge Morales in the following paper:
Jorge Morales. The classification of pairs of binary quadratic forms. Acta Arith., 59(2):105–121, 1991.
In ...
- 21.4k
4
votes
On the equivalence of a pair of binary quadratic forms
We can show that $ h(1,1,n) $ increases like $ n $-th powers of 2 : indeed, in the half-plane presentation of the set of positive definite quadratic forms, the euclidean form is identified to $ ...
- 1,960
4
votes
On the notion of primary representation of a natural number by a quadratic form
I'm going to take a more algebraic number theoretic approach to this question. The solutions to $x^2-dy^2=k$ are the integers of norm $k$ in the number field $\mathbb{Z}[\sqrt(d)]$ if $d$ is a ...
- 2,196
4
votes
Is the set of integers represented by a quadratic form of non-fundamental discriminant a subset of the rep. set of a form of fundamental discriminant?
This is true, of course. The book you want is Binary Quadratic Forms by D. A. Buell. He talks about how, given some discriminant $\Delta$ and class number(primitive forms) $h(\Delta),$ we can predict $...
- 24.5k
3
votes
Accepted
Distribution of 'square classes' of binary quadratic forms
The standard definition for a form to belong to the principal genus of forms with fundamental discriminant $d$ is that the primes $p$ coprime to $d$ that
the form $Q$ represents satisfy $(d_1/p) = \...
- 30.6k
3
votes
Is the diophantine equation $3x^2+1=py^2$ always solvable for each prime $p\equiv 13\pmod{24}$?
Here's a second proof of Claim 1 (I guess the others can be taken care of similarly) that perhaps explains a little bit better what is going on. Assume first that $3x^2 + 1 = py^2$ has an integral ...
- 30.6k
3
votes
Classifying binary quadratic forms by the set of representable squares
No, the set $S_f$ is empty for lots of forms. For example, if $a$ is not a square modulo $b$, then $ax^2+by^2$ is never a square (consider the equation modulo $b$). Thus, $S_f$ is empty for $f(x,y)=6x^...
- 12.2k
3
votes
Accepted
Lower bound on class number of binary quadratic forms of discriminant of the form $n^2+4$
see Mollin's paper and the various references given there. You will probably also find relevant
material in his book "Quadratics".
- 30.6k
3
votes
Representation of integers by principal binary quadratic forms
As noted in an earlier answer referring to Pall's 1969 paper in the Journal of Number Theory, it is unlikely that there is a simple way to describe all the divisors of $k$ that are represented by the ...
- 18.7k
3
votes
Accepted
Clarification regarding a claim in Heilbronn’s 1934 paper
One can see more directly that $a^H$ has a representation by the principal form with $y\neq 0$ (hence also with $y>0$). Indeed, let
$$\omega:=\begin{cases}\sqrt{d},&d\equiv 0\pmod{4}; \\
(1+\...
- 85.6k
2
votes
Does the Diophantine equation $(x^2+ay^2)(u^2+bv^2) = p^2+cq^2$ admit a complete solution?
I don't think it has been explicitly mentioned yet, but presumably you are interested in integer solutions for given values of $a, b, c$.
L E Dickson's "History of the Theory of Numbers" vol 2 [ http:...
- 1,469
Only top scored, non community-wiki answers of a minimum length are eligible