# Tag Info

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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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### 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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### 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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### 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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### 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 ...

### $x^2+7y^2=2^n$ 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 ...

### 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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### 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, ...

### 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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### 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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### 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 ...

### 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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### 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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### 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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### 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 $\langle a,b,c \rangle$ ...

### 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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### 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, ...
### 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:...