**2**

votes

**1**answer

88 views

### Set of triple-primes satisfying a certain equation

Is there a set of triple-primes satisfying the following equation?
$p_1p_2+p_2p_3+p_3p_1+p_1+p_2+p_3=2^β,p_1p_2p_3=2^α-1,α>β.$
I have checked the first 11 numbers that no one satisfy the above ...

**12**

votes

**1**answer

212 views

### A prime ideal $\mathfrak{p}$ decomposes in $\mathbb{Q}(\zeta_{24})/\mathbb{Q}(\sqrt{-6})$ iff it is generated by $\alpha\in1+2\Bbb{Z}[\sqrt{-6}]$

For a nonzero prime ideal $\mathfrak{p}$ of $\mathbb{Z}[\sqrt{-6}]$ which does not divide $2$, does $\mathfrak{p}$ decompose completely in the extension $\mathbb{Q}(\zeta_{24})/\mathbb{Q}(\sqrt{-6})$ ...

**8**

votes

**1**answer

259 views

### Have the explicit Poisson-type formulas of Guinand and Meyer been observed before?

In a recent paper of Meyer Measures with locally finite support and spectrum PNAS vol. 113 no. 12:3152–3158 (behind a paywall, but see also these seminar notes) some new explicit Poisson-type formulas ...

**8**

votes

**0**answers

242 views

### Has it been proved that odd perfect numbers cannot be triangular?

(Note: This question has been cross-posted from MSE.)
Euclid and Euler proved that every even perfect number is of the form $m = \frac{{M_p}\left(M_p + 1\right)}{2}$ where $M_p = 2^p - 1$ is a prime ...

**18**

votes

**1**answer

302 views

### On random divisor sums modulo $2^k$

Let $k,n,\ell$ be positive integers with $k,n\ge 2$ and $0\le \ell \le k-1$. For each integer $2\le j \le n$, choose a divisor $d_j$ of $j$, uniformly at random from the divisors of $j$. We denote by ...

**2**

votes

**0**answers

106 views

### Extension to real number system [closed]

Suppose you have equation involving a number $s$
$s^2+ 1 = 0$,
to solve it one needs to treat $s$ as complex number $s = \pm i$, and introduce $i$ as imaginary unit.
Now suppose you have equation ...

**7**

votes

**0**answers

131 views

### k-Almost Primes in short intervals

According to this question every interval $[x, x + x^{0.45}]$ contains a product of two primes, and this has been improved further slightly. Are there better results available for $k$-almost primes? ...

**21**

votes

**2**answers

514 views

### CM $j$-invariants in $p$-adic fields

I'm trying to understand the $p$-adic distribution of $j$-invariants for elliptic curves with complex multiplication.
Specifically, suppose $\sigma$ is some embedding $\sigma:\overline{\mathbb Q}\to ...

**23**

votes

**1**answer

492 views

### Artin reciprocity $\implies $ Cubic reciprocity

I asked this on math.SE a few days ago with no reply, so I'm reposting it here. Hope this is not considered too elementary for MO (feel free to close if so).
I'm trying to understand the proof of ...

**2**

votes

**0**answers

59 views

### Property of a derivative in global field

Before posting I want to make it clear that I posted the same question in stack exchange awhile ago (http://math.stackexchange.com/questions/1533814/property-of-derivative-in-a-local-field) but didn't ...

**6**

votes

**1**answer

225 views

### Can $b^4+1$ be a pseudoprime to base 2 (except for Fermat numbers)?

Cross-post: This very elementary question was first posted to Mathematics Stack Exchange but the response I got there (even after offering a bounty) was not useful.
For the purpose of this question, ...

**3**

votes

**0**answers

74 views

### Sign of bivariate polynomial evaluated over two algebraic numbers

I would like to compute the sign of a bivariate polynomial $f$ evaluated over two algebraic numbers $a$, $b$. The numbers are in "isolating interval representation" meaning that each one is defined by ...

**12**

votes

**0**answers

380 views

### Elkies' supersingularity theorem in higher dimension (in terms of the associated Newton polygon)

Elkies' supersingularity theorem: Given an elliptic curve $E$ over $\mathbb{Q}$, there are infinitely many primes $p$ such that $E$ is supersingular over $\mathbb{F}_p$.
I have seen another post on ...

**6**

votes

**1**answer

137 views

### What is the cokernel of $O_S \to F_\infty/O_\infty$?

Let $k$ be a field of characteristic $\neq 2$ and consider the quadratic extension $F$ of $k(T)$ generated by $\sqrt{T^3 + 1}$. Let $X$ be the set of all places of $F$. Let $S = \{\infty\} \subset X$ ...

**5**

votes

**1**answer

199 views

### Bounding $p$-adic characters and Jacquet-Langlands tranfert

I would like to bound uniformly in $\pi$ the $p$-adic Harisch-Chandra characters $\Theta_p$ for division quaternion algebras. By the Jacquet-Langlands correspondence, it is sufficient to bound it on ...

**3**

votes

**1**answer

315 views

### Relation between the binary Goldbach problem and binary version of Mobius sum

What I want to ask is about the structure of the Goldbach function that defined by
$$ R(x)=\#\{ p \mid x-p \in \mathbb{P} , \ p\leq x/2\}$$
for $x\in 2\mathbb{N}$, where $\mathbb{P}$ is the set of ...

**2**

votes

**2**answers

289 views

### A class of quadratic equations

Let $f(x,y) = ax^2 + bxy + cy^2$ be an indefinite irreducible binary quadratic form with integer coefficients with non-zero discriminant. We can assume, without loss of generality, that $a \geq 1$. ...

**35**

votes

**3**answers

3k views

### Why such an interest in studying prime gaps?

Prime gaps studies seems to be one of the most fertile topics in analytic number theory, for long and in lots of directions :
lower bounds (recent works by Maynard, Tao et al. [1])
upper bounds ...

**1**

vote

**0**answers

198 views

### On a sequence of L-functions having same zeros in critical strip and GRH

I had an idea on GRH involving a sequence of L-functions having same zeros, then at one step I need a bound on these function and I wonder if this bound is in fact not as hard as GRH itself ?
Let's ...

**2**

votes

**0**answers

69 views

### the least point on a variety over a finite field

Let $p$ be a large prime parameter and $V\subseteq \mathbb{P}^n_{\mathbb{F}_p}$ a variety defined over the finite field $\mathbb{F}_p$ with bounded degree and dimension (w.r.t. $p$). Assume that $V$ ...

**0**

votes

**1**answer

230 views

### On elliptic curves, $\sqrt{x^2-101y^2} ,\sqrt{x^2+101y^2}$, and their ilk

I. Elliptic curves
Given integers $a,b,m_k$. Let,
$$x^2+a = m_1u_1^2\\x^2+b = m_1u_2^2\tag1$$
If there is a rational point $x_i$, then the pair (after a transformation) is birationally ...

**0**

votes

**0**answers

98 views

### Abel summation formula versus Perron's formula to bound a partial sum

Taking $\chi$ a primitive character, with Abel summation it is easy to show that for $\epsilon >0$, there is a constant $M$ such that for all $x$ we have :
$$|\sum_{n<x} ...

**2**

votes

**0**answers

92 views

### A reference about a problem of the number of the rational points on a projective scheme

Let $X\hookrightarrow\mathbb{P}^n_{\mathbb{F}_q}$ be a pure dimensional projective scheme of dimension $d$. So we know a trivial estimate of the number of $X(\mathbb F_q)$ is that $\#X(\mathbb ...

**0**

votes

**1**answer

313 views

### How do Modular Forms and the Geodesic Flow interact? [closed]

Textbooks talk at length of the modular properties of $\theta(z)$ or $\tau(z)$ and the prominent role of $SL(2,\mathbb{Z})$ or one of the congruence groups.
In that case, aren't the basic objects ...

**4**

votes

**3**answers

391 views

### On a theorem of Hensel about congruence of binomial coefficient

In the paper Binomial coefficients modulo prime powers, Andrew Granville stated the following theorem:
Let $n, m$ and $r=n-m$ be three given positive integer and $p^k$ is the exact power of $p$ ...

**3**

votes

**1**answer

72 views

### Finiteness of the $p$-primary subgroup of an elliptic curve over the cyclotomic $\mathbb{Z}_p$-extension

Let $E$ be an elliptic curve defined over a number field $F$ and $F_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of $F$. Is it true that the $p$-primary subgroup of $E$ over $F_\infty$ i.e. ...

**8**

votes

**1**answer

323 views

### Is there any real quadratic ring for which the Euclidean algorithm is polynomial?

We know from Rolletschek's work that the Euclidean algorithm of $\mathbb{Z}[i]$ is polynomial. Indeed, let $n$ be the maximum number of steps in the Euclidean algorithm applied to $u,v ...

**4**

votes

**0**answers

136 views

### extending $p$-adic character of the local intertia to the absolute Galois group

Suppose I have a number field $F$, and a finite place $v$ of $F$. Let $E$ be finite extension of $F_v$. I start with a continuous morphism
$$
\chi \colon O_{F_v}^\times \to E^\times.
$$
where ...

**7**

votes

**0**answers

190 views

### Can primes be (almost) random sequence in von Mises sense?

Random models for primes (such as Cramer's model) have been extensively used for informal justification of various conjectures involving primes. It is crucial to understand in what sense sequence of ...

**20**

votes

**3**answers

740 views

### Does X(13) have potentially good reduction at 13?

The complete level modular curve $X(p)$ does not have potentially good reduction at $p$ for any $p \neq 2,3,5,7,13$ because then there are cusp forms on $X_0(p)$ showing up in the cohomology of ...

**0**

votes

**1**answer

202 views

### Is it true that singular fibers of elliptic fibrations that have the same Kodaira type are isomorphic schemes?

It is well known that Kodaira gave an essentially topological classification of the possible singular fibers of elliptic fibrations according to their type:
...

**8**

votes

**2**answers

1k views

### Approximating any integer by multiples of 2 and 3

Given any integer $n$ sufficiently large, I want to prove (or disprove) that there exists another integer $m\ge n$ with the form $m=2^a3^b$ ($a,b$ are no negative integers) such that $m-n=o(n)$, i.e., ...

**4**

votes

**1**answer

189 views

### Eisenstein Series on Siegel Space

I am looking for any reference dealing explicitly with Eisenstein series on Siegel space (the simplest case of $\rm{SP}_4$ is fine). Anything would be welcome, but in particular I'm interested in the ...

**0**

votes

**1**answer

137 views

### An upper bound on $\sum_{n^{1/3}<p,q\leq n^{1/2}} \frac{n}{pq}-\lfloor \frac{n}{pq}\rfloor$

I would like to ask if there is a good upper bound on the difference $$D_2(n)=\sum_{n^{1/3}<p,q\leq n^{1/2}} \left(\frac{n}{pq}-\left\lfloor \frac{n}{pq}\right\rfloor\right)\quad (1) $$where $p$ ...

**5**

votes

**1**answer

166 views

### Counting the number of permutations of $(1,\ldots,i,\ldots,j,\ldots,m)$, where $i < j$ and number of inversions is $k$

How can I prove the following:
$d^{ij}(m,k) > d^{ji}(m,k)$ for all $k < \frac{1}{2}\binom{m}{2},$
where $d^{ij}(m,k)$ denotes the number of permutations of $(1,\ldots,i,\ldots,j,\ldots,m)$ ...

**0**

votes

**2**answers

508 views

### Is the absolute Galois group the same as the automorphism group? [closed]

Is the absolute Galois group $\mathrm{Gal}(\overline{\mathbf{Q}}|\mathbf{Q})$ the same as the group $\mathrm{Aut}_{\mathbf{Q}}(\overline{\mathbf{Q}})$ the automorphism group in the category of ...

**3**

votes

**1**answer

312 views

### Cohomology of elliptic curves

Assume $K$ is an imaginary quadratic extension of $\mathbb{Q}$, and $E$ an elliptic curve defined over $\mathbb{Q}$.
Let $p\neq l$ be primes in $\mathbb{Q}$ where $E$ has good reduction. Assume $p$ ...

**15**

votes

**2**answers

2k views

### Could this unexpected bias in the distribution of consecutive primes have any impact on the security of encryption algorithms?

In a recent paper a quite unexpected result about a new pattern in prime numbers emerged:
Unexpected biases in the distribution of consecutive primesby Oliver, R. J. L.; Soundararajan, K. (Submitted ...

**9**

votes

**1**answer

448 views

### combinatorics on cyclic sequences

Given $m\geq 1$, let $I=(a_1,\ldots,a_{3m})$ be a sequence such that $I$ contains exactly $m$ zeros, $m$ ones, and $m$ twos.
Given $i=1,2$ and $j\leq 3m,k\leq m$ we can define ...

**5**

votes

**3**answers

439 views

### Extending rational Diophantine triples to sextuples

(This is a follow-up to a previous post.) A rational Diophantine $m$-tuple is a set of rationals {$a_1,a_2,\dots a_m$} such that (with $i\neq j$), all $a_i a_j+1$ is a square. Problem: Find a class of ...

**2**

votes

**0**answers

72 views

### Transcendence of a $q$-series

Let $q\ge 2$ be an integer. Fourier's proof of irrationality of $e$ adapts to prove the irrationality of
$$\Psi_q=\sum_{n\ge0}\frac1{\prod_{k=0}^{n-1}(q^n-q^k)}$$
Is this number knwon to be ...

**4**

votes

**1**answer

139 views

### Lifting of Frobenius on torsors over abelian varieties

This is related to my previous question Assume that $A$ is an abelian variety over a field $k$ of characteristic $p$, $\mathcal{L}$ is a line bundle on $A$. Assume that $A$ is ordinary and ...

**3**

votes

**1**answer

273 views

### Generalizing a pattern for the Diophantine $m$-tuples problem?

A set of $m$ non-zero rationals {$a_1, a_2, ... , a_m$} is called a rational Diophantine $m$-tuple if $a_i a_j+1$ is a square. It turns out an $m$-tuple can be extended to $m+2$ if it has certain ...

**2**

votes

**0**answers

137 views

### Bounds for an Egyptian Fraction Inequality

Question: If $A\geq B>0$ are rational and $x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ are integers such that $A\geq \sum_{j=1}^{n}\frac{1}{x_{j}}\geq B$, then what is an upper bound on $x_{j}$ in terms ...

**5**

votes

**0**answers

108 views

### Factorization problem in Cyclic cubic field

Let K/$\mathbb{Q}$ be a cubic number field. Assume that K/Q be Galois with class number 1.
Therefore Gal(K/Q) is cyclic cubic group and $\mathcal{O}_K$ is a PID.
Let p be a rational prime, p ...

**22**

votes

**1**answer

463 views

### Why is there a Parity Problem in Sieve Theory and not a Mod p problem for any other p?

The "parity problem" in sieve theory, so far as I understand it, is the fact that sieves can't distinguish between primes and $2$-almost primes, numbers with exactly two prime factors, and will always ...

**3**

votes

**0**answers

169 views

### Repartition of 1's in the “Chacon word”

Consider the "Chacon words": $B_0=0$ and $B_{n+1} = B_nB_n1B_n$. The word $B_n$ has $\ell_n := \frac{3^{n+1}-1}{2}$ digits and the number of $1$'s in $B_n$ is $\ell_n - 3^n = \ell_{n-1} \sim ...

**4**

votes

**0**answers

206 views

### Asymptotic estimate for a random model of primes

Question
Let
$$
\pi_{rm_c}(x) = \sum_{ \substack{ {n\leq x}\\{(n+a,P(\sqrt{n}))=1}}} 1-1,
$$
where $P(x)$ is the product of all primes less or equal to $x$ and $a$ is a random integer constrained to ...

**6**

votes

**2**answers

330 views

### Are the abelian absolute Galois groups of these local fields isomorphic?

For a field $F$ we denote by $F^{\mathrm{ab}}$ the compositum of all finite Galois abelian extensions of $F$.
Is $\mathrm{Gal}(\mathbb{Q}_2(\sqrt[8]{3})^{\mathrm{ab}}/\mathbb{Q}_2(\sqrt[8]{3})) ...

**18**

votes

**1**answer

709 views

### Concrete Applications of knowing $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$

I have very little experience with Galois representations, mostly as they relate to class field theory, elliptic curves, and modular forms, but they seem to have quite a reputation in number theory as ...