20 votes
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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 ...
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20 votes
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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 ...
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19 votes
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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 ...
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18 votes
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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 ...
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16 votes
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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 ...
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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 ...
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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 ...
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11 votes
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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, ...
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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 ...
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10 votes
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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 \...
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10 votes
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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 ...
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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 ...
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8 votes
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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 ...
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8 votes
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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....
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8 votes
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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 $\langle a,b,c \rangle$ ...
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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 $-a < b \leq a < \sqrt{- D /3 }$. ...
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5 votes
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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, ...
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5 votes
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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) := \...
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4 votes
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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 ...
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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 ...
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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 $ ...
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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 ...
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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 $...
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3 votes
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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) = \...
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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 ...
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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^...
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3 votes
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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".
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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 ...
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3 votes
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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+\...
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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:...
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