Prime numbers, Diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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37 views

Algorithm to compute a common denominator of a finite set of rational numbers

Let $x_1,\dots,x_n \in \mathbb{Q} \cap (0,1)$ be fixed, but unknown, and assume that we know a number $K \in \mathbb{N}$ such that there is a number $N \in \{1,\dots,K\}$ such that $x_1 N,\dots,x_n N ...
0
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0answers
8 views

$B_k[1]$ sets with smallest possible $m = max B_k[1]$ for given $k$ and $n = |B_k[1]|$ elements

Sidon sets are sets $A \subset \mathbb{N}$ such that for all $a_j,b_j \in A$ holds $$a_1+a_2=b_1+b_2 \iff \{a_1,a_2\}=\{b_1,b_2\}$$ Thus if you know the sum of two elements, you know which elements ...
5
votes
0answers
105 views

Spencer's “six standard deviations” theorem - better constants?

This question is about Joel Spencer's famous "six standard deviations" theorem. If you don't know the theorem, it's Theorem 1 in: Spencer, Joel. Six standard deviations suffice. Trans. Amer. Math. ...
2
votes
1answer
102 views

Regarding the set up of a geometry of numbers lemma

I have a question related to geometry of numbers, which although seems quite basic, I was rather confused by it so I decided to ask here. Let $\Lambda$ be a lattice in $\mathbb{R}^n$. Let $R_1, ..., ...
6
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0answers
112 views

Power series defined by Witt vectors / Teichmüller representatives of p-adics

Let $K$ be $\mathbb{Q}_p$ for some prime $p$ (or more generally an unramified extension $W(\mathbb{F}_q)$ of $\mathbb{Q}_p$). If $\xi \in K$, we can write it in a unique way in the form $\sum a_i ...
1
vote
0answers
151 views

Number of Orbits of symmetric group acting on $(\mathbb{Z}/n)^{l}$ [migrated]

I have encountered a problem that I suspect has been thoroughly studied but I have not been able to find references. Can anyone point me to a published reference dealing with this or a closely related ...
2
votes
0answers
81 views

Bloch Kato Exponential as formal lie group exponential

Let $K$ be a $p$-adic field and $V$ a $p$-adic representation. In their paper on tamagawa numbers of motives, Bloch and Kato define an exponential map as the connecting homomorphism $$DR(V) ...
1
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0answers
109 views

Arguments for the second Hardy–Littlewood conjecture being false?

Assume that $x,y > 2$, and that $x<y$. Then the second Hardy–Littlewood conjecture states that $$\pi(x + y) - \pi(y) \leq \pi(x).$$ We can easily justify this heuristically, since $$ ...
2
votes
0answers
51 views

Asymptotic formula for restricted partition function

Let $p(n)$ be the partition function. Hardy and Ramanujan - and Uspensky, independently proved the asymptotic formula $$(1) \quad p(n) \sim \frac1{4\sqrt{3}} \frac{e^{c_0\sqrt{n}}}{n} \text{ as } n ...
17
votes
1answer
513 views

The conjecture of Montgomery and Soundararajan on primes in short intervals: Empirical inconsistencies?

Assume that $y/ \log x \rightarrow \infty$ and that $y/x \rightarrow 0$. Then, from a conjecture by Montgomery and Soundararajan, we expect the number of primes in the interval $[x,x+y]$ to be ...
3
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0answers
75 views

Arithmetic analogs of vertex algebras?

Has anyone successfully defined and studied analogs of vertex algebras where the grading of the fields is by $(\log \mathbb Q)$ rather than $\mathbb Z$? What I mean is that the usual fields $$ a(z) = ...
0
votes
0answers
113 views

Koblitz - Are chapters III & IV independent of I & II [on hold]

I am interested in learning about modular forms and have heard many great things about Neal Koblitz's Introduction to Elliptic Curves and Modular Forms. However, Koblitz doesn't discuss modular forms ...
1
vote
0answers
210 views

Proportion of rational elliptic curves of a given rank

This morning appeared on Arxiv the following article by Manjul Bhargava et al: http://arxiv.org/pdf/1407.1826.pdf, in which the authors give a lower bound for th proportion of rational elliptic curves ...
2
votes
1answer
140 views

Extension of a formula for the quadratic Gauss sums

I am interested in the sums $$g(a,k)=\sum_{n=0}^{p-1}e^{2\pi i a n^k/p}$$ where $p\equiv1\mod k$ is a prime and $a$ is coprime with $p$. When $k=2$, it is a classical fact that $g(a,2)=\chi(a)g(1,2)$ ...
2
votes
0answers
50 views

Base change of discrete series

Let $\pi_f$ be an automorphic representation of $GL_2(A_Q)$ (attached to a modular form $f$), and suppose we want to look at its base change lift to say a quadratic imaginary field. Which are the ...
2
votes
1answer
97 views

About the restriction of a modular representation to a decomposition subgroup II

This question is a variant of this one. Let $f$ be as in the other question, but suppose that we look at the $\ell$-adic representiation attached to $f$: $$ \rho_f : G_{\mathbb Q} \to ...
3
votes
1answer
118 views

Regular singularities and the infinitesimal site

Suppose I have a smooth non-proper algebraic variety $X/\mathbb{C}$. A vector bundle with flat connection (``differential equation'') on $X$ extends, as was noted by Grothendieck, to a coherent ...
0
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0answers
203 views

Arithmetic progression and average of two prime numbers

Let $A=(a_n : n \in \mathbb{N})$ be the sequence given by: $$ \ a_n = a_1 + (n - 1)d,\quad a_1,\ d,\ n \in \mathbb N,\quad d\gt a_1,\quad \gcd(a_1,\ d)=1. $$ For all terms of $A$ greater than $\ ...
-4
votes
0answers
27 views

calculating value of higher powers [on hold]

calculating the remainder of higher powers eg : 2008 ^2008 /9 . How to Proceed with this . Is there any shortcut methods or we should be doing this with the cyclicity principle .
0
votes
1answer
72 views

rank of Abelian schemes under ample hypersurface section

Let $k$ be an algebraic closure of a finite field, $\ell \neq \mathrm{Char}(k)$ be prime, $S/k$ a smooth projective geometrically connected surface and $C/k$ a smooth ample connected hypersurface ...
2
votes
1answer
149 views

About the restriction of a modular representation to a decomposition subgroup

Let $f$ be an eigenform of level $\Gamma_1(N)$ and let $p$ be a prime that does not divide $N$. It is well know that there is a $2$-dimensional representation $$ \rho_f \colon G_{\mathbb Q} \to ...
2
votes
1answer
187 views

Proof of the Friedlander–Iwaniec theorem

Does anybody know where I could find the proof of the Friedlander–Iwaniec theorem. The link that I find when I search for it is http://www.pnas.org/content/94/4/1054.full.pdf+html, but this seems more ...
17
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0answers
453 views

Base change for $\sqrt{2}.$

This is a direct follow-up to Conjecture on irrational algebraic numbers. Take the decimal expansion for $\sqrt{2},$ but now think of it as the base $11$ expansion of some number $\theta_{11}.$ Is ...
-2
votes
1answer
129 views

Decimal digits multiplied by powers of 2: leads to mod 8? [on hold]

This is more a puzzle than a research question, a puzzle to me. Perhaps it is straightforward for others. Imagine Repeatedly interpreting a number expressed with the usual base-$10$ digits as ...
0
votes
1answer
270 views

How can I calculate $ \sum_{i=1}^{n} (n \bmod i) $

I want to calculate the sum $$ \sum_{i=1}^{n} \left\lfloor\frac{n}{i}\right\rfloor, $$ and it seems to require to calculate the sum $$ \sum_{i=1}^{n} (n \bmod i). $$ How can I get an $O(1)$ ...
3
votes
0answers
88 views

Equidistribution of double coset

Let $G=PGL_n(\mathbb{R})$, $K=PO_n(\mathbb{R})$ and $X=G/K$. Also suppose $\Gamma=SL_n(\mathbb{Z})$ acts on the left of $X$. We define a typical Hecke operator on $L^2(\Gamma\backslash X)$ by the ...
9
votes
2answers
257 views

how to find cubic polynomial that an unknown subset of a set of integers satisfies

I have a set, $S$, of positive integers and I have reason to believe that some infinite subset of them may be parametrized by a cubic polynomial with integer coefficients evaluated at integer ...
0
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0answers
92 views

Vanishing sum of roots of unity with fixed weight

Consider the set $S_n(d) = \{(x_1, \cdots ,x_n) \in \mathbb{C}^n \mid 1 + x_1+\cdots +x_n = 0, x_i^d=1 \text{ for all } i=1,\dots,n.\}$ Many papers which I found consider the conditions on $n$ and ...
1
vote
2answers
268 views

Non-hyperelliptic curves of genus at least two

A hyperelliptic curve can be understood as the set of points satisfying an equation of the form $$\displaystyle z^2 = f(x,y),$$ where $f(x,y)$ is a binary form of degree $d = 2g+2$. In this case, $g$ ...
5
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0answers
150 views

Which L-functions are not “Langlands-Shahidi L-functions”?

The Langlands-Shahidi method, among other things, obtains certain L-functions from the constant term of Eisenstein series attached to so-called $(G,M)$ pairs, where $G$ is a reductive group, $M$ a ...
1
vote
1answer
280 views

What is the value of $\sum _{n=1}^{\infty \:}\frac{n!}{n^n}$? [closed]

I have a question: What is the value of $\sum _{n=1}^{\infty \:}\frac{n!}{n^n}$? Only I've calculated the following identity: $$\sum _{n=1}^{\infty \:}\frac{n!}{n^n}=\int _0^{1}\left(1+x\cdot \ln ...
3
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1answer
348 views

Orders of Finite Simple Groups

Which finite simple groups have order N so that N+1 is a proper power? As an example: the simple group of order $168=13^2-1$.
2
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3answers
301 views

Equivalent binary forms

Two binary forms $f, g \in k[x, y]$ are equivalent when there exists an $M \in GL_2 (k)$ such that $f^M = g$. For simplicity we take $k$ such that $char (k) =0$ and $k=\bar k$. The equivalence ...
1
vote
1answer
162 views

What can be said about the asymptotics of this sequence?

This is a more general version of a question I have asked before. Let $a_1,\dots,a_r\in\mathbb C$ be algebraic numbers and suppose that for every natural number $n$ the sum $$ F(n)=\sum_{j=1}^r a_j^n ...
-1
votes
0answers
47 views

Congruences of exponential functions [closed]

I am looking for books and papers which address the topic of exponential functions mod m, in particular I am interested in the periodicity of the function 2^x under moduli which are powers of 3. How ...
-3
votes
0answers
130 views

Can anyone extend my findings about Langford Pairings? [closed]

$n= 3; N = \large \frac{1*(2n)!}{(2n -1)! *n} - \frac{(2^2 -n -1) *}{} = 2$; last fraction is zero. $n= 4; N = \large\frac{1*(n-3)*(2n)!}{(2n -1)! *n} - \frac{(2^3 -2*n)*}{} = 2$; last fraction is ...
-2
votes
0answers
67 views

Action of the group Z_p on a module M [closed]

I have a very stupid but elementary question. Assume that the group ${\Bbb Z}_p$ acts on a module $M$. Assume that the topological generator $\sigma (=1)$ of ${\Bbb Z}_p$ acts trivially on an element ...
0
votes
0answers
258 views

Are these roots of unity?

Suppose you are given complex numbers $a_1,\dots,a_N$ which are algebraic over $\mathbb Q$ and satisfy $1=|a_1|=\dots=|a_k|>|a_{k+1}|\ge\dots\ge|a_N|$. Assume that $$ \sum_{j=1}^Na_j^m $$ is a ...
-4
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0answers
66 views

Finding out the remainder [closed]

I need to find the remainder when 20!+20^23 is divided by 23 . .please help. Wilson theorem involved , Fermat's little theorem involved as well.
9
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2answers
439 views

Bound on gcd of two integers

Well this is a problem I was fiddling with. I came up with it but it probably is not original. Suppose $a\in \mathbb{N}$ is not a perfect square. Then show that : ...
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2answers
322 views

Chinese Remainder Theorem backwards

I have the following situation, that is much alike the Chinese Remainder Theorem. Let $\phi_d(\alpha)$ be the $d^{th}$ cyclotomic polynomial in the variable $\alpha$ (I'm not specifying the ...
2
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0answers
212 views

schanuel's conjecture and real root of $x+e^x=0$

Schanuel's conjecture states: -If $\alpha_1,\alpha_2,...,\alpha_n$ are complex numbers linearly independent over $\mathbb{Q}$ then the trascendence degree of the field ...
2
votes
0answers
68 views

Primality Criterion for Specific Class of Numbers of the Form kb^n-1

Let $N=k\cdot b^n-1$ where $b$ is an even integer , $3\nmid b$ , $3\nmid N$ , $k \equiv 1,5 \pmod{6}$ , $k< b^n $ and $n>2$ . Let $S_i=P_b(S_{i-1})$ with $S_0=P_{k\cdot b/2}(P_{b/2}(4))$ , ...
0
votes
0answers
87 views

Solutions to Diophantine equation [closed]

$$\displaylines{a = (x-y)b - xy\cr 0\le x, y \le b^2\cr x, y \in\ {\bf N}\cr a > 0 \in\ {\bf N}, b > 0 \in\ {\bf N}\cr} $$ Is this diophantine equation studied in the literature? Or is there a ...
0
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0answers
14 views

Points in a general Cantor set [migrated]

We often look at the Cantor set with the construction that keeps removing the middle thirds of the remaining line segments at each iteration. Corresponding to this construction, we can determine ...
4
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0answers
91 views

Complexes of arithmetic $\mathcal{D}$-modules with Frobenius structure

This is a question about the category $F\text{-}D^b_\mathrm{coh}(\mathscr{D}^\dagger_{\mathscr{X},\mathbb{Q}})$ of complexes of arithmetic $\mathscr{D}$-modules with Frobenius structure on a smooth ...
16
votes
1answer
342 views

Covering a set with geometric progressions

Consider the set $S_n=\{1,2,\cdots ,n\}$. What is the minimum number of distinct geometric progressions that cover $S_n$? Let us call this number $a_n$. I was wondering about this number after doing a ...
2
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0answers
69 views

Berkovich Analytification of the transseries

I am looking for references to articles about the following subjects: Connections from the field of (real) transseries to the field of surreal numbers (mentioned very briefly in the introduction of ...
2
votes
0answers
79 views

Primality Criterion for Specific Classes of Generalized Fermat Numbers

Let $F_n(b)= b^{2^n}+1 $ where $b$ is an even integer , $ 3\nmid b , 5\nmid b $ and $n\ge2$ Let $S_i=P_b(S_{i-1})$ with $S_0=P_{b/2}(P_{b/2}(8))$ where ...
8
votes
0answers
133 views

Symmetric Fifth Power Lift of GL(2) Automorphic Form

Let $\pi$ be an automorphic representation of $GL(2)/\mathbb{Q}$. For simplicity, you can take it to be a Maass form for $SL(2,\mathbb Z)$. Kim, Shahidi, Gelbart-Jacquet prove that $$L(s, \pi, ...