# Tagged Questions

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87 views

### Any result, example or conjecture about the computational complexity of transcendental number that can not be computed by linear time Turing Machine [on hold]

The Hartmanis-Stearns Conjecture is restated by Prof.Lipton as:
Suppose that a linear time Turing Machine computes the first $n$ digits of the real number $r$ in base ten. Then, the number is either a ...

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64 views

### Are there any relevance between coefficients of simple continued fraction of quadradic algebraic number and algebraic number with degree $2^n$ [on hold]

Let $\sqrt{c}$ be quadratic algebraic number, We know that $[a_0;a_1,a_2,\dots ]$ the coefficients of simple continued fraction of $\sqrt{c}$ the quadratic algebraic number is periodical. ...

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80 views

### What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers $\sqrt{A}$ and the integer $A$

What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers and the integer
As we know,$\sqrt{2} = [1;2,2,2,2,…]$; while $\sqrt{14}= ...

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**0**answers

198 views

### What does a Turing machine compute? [on hold]

I suspect that it might be necessary to define for a Turing machine how its inputs and outputs are to be interpreted in order to be able to say e.g. that a Turing machine $T$ computes an arithmetical ...

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23 views

### Proof that $G(3)\le 7$ [migrated]

Let $G(k)$ be the minimal $n$ s.t. every sufficiently large integer is the sum of $n$ nonnegative $k$th powers.
Does anybody know where I can find Vaughan's proof that $G(3)\le 7$? I can't find a ...

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86 views

### Collatz property implying infinite “fall below” trajectories, is it known?

(this was discovered analyzing Collatz empirically.)
a key aspect of resolving Collatz involves looking at the number of iterations for trajectories to "fall below" the initial value.
consider ...

**8**

votes

**1**answer

209 views

### Integers with a large prime divisor in short intervals

For an integer $n$, denote by $P^+(n)$ the largest prime divisor of $n$. Then we have the following:
There exists some $c>0$, such that for all $x$ sufficiently large the number of integers ...

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55 views

### Property of set of prime numbers [migrated]

let $\{p_1,p_2,p_2,\cdots ,p_r\}$ be the set of $r$($\ge2$) pair wise distinct prime numbers
i.e.., $(i\ne j \implies p_i \ne p_j)$ for all $1\le i,j\le r$
${Statement}$ :
For any such ...

**-1**

votes

**1**answer

131 views

### Conjectured Primality Test for Numbers of the Form k2^n+1 with n>2 [closed]

Definition : Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^m+\left(x+\sqrt{x^2-4}\right)^m\right) $
where $m$ and $x$ are positive integers .
Conjecture : Let $N=k\cdot 2^n+1$ with ...

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votes

**1**answer

168 views

### Atkin--Lehner operators in Hida theory

Let $p$ be a prime, and $F$ a $p$-adic Hida family of ordinary modular forms (of some tame level $N \ge 1$). I'd like to know whether, for $q$ a prime factor of $N$, the actions of the Atkin--Lehner ...

**8**

votes

**1**answer

428 views

### Rank of Elliptic Curves

Recently, I have heard of some heuristics that would suggest that the rank of elliptic curves are bounded (specifically in the congruent number family). I always though that the best way to prove ...

**14**

votes

**1**answer

728 views

### Is $x^{n}-x-1$ irreducible?

Is it true that for every $n \in \mathbb{N}$, $x^{n}-x-1$ is irreducible in $\mathbb{Z}[x]$?
The standard irreducibility criteria seem to fail.

**4**

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**1**answer

473 views

### 3n+1 problem and cycles

Just to make sure I am up to date with this problem. I know (or I think I do) that it is not yet proven that there are no non-trivial cycles for the collatz sequence (please correct me if I am wrong). ...

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**3**answers

1k views

### Is any particular algebraic number known to have unbounded continued fraction coefficients?

The continued fraction
$$[1;1,2,3,4,5,\dots]=1+\cfrac{1}{1+\cfrac{1}{2+\cdots}}, $$ for instance, is known explicitly as a ratio of Bessel function values and is (I believe - SS) known to be ...

**4**

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**2**answers

333 views

### Are there any patterns in simple continued fraction expansions of algebraic real numbers?

As we know there are patterns in simple continued fraction expansion of quadratic algebraic numbers,are there any patterns in simple continued fraction expansions of other algebraic real ...

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112 views

### Asymptotics and error terms for an arithmetic function built upon $\omega$ and $\Omega$ functions

For any real number $x$, let's define $Om_{k}(x)$ as the number of positive integers $m$ below $x$ such that $\Omega(m)-\omega(m)=k$, where $\omega(n)$ is the number of distinct primes dividing $n$, ...

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**3**answers

1k views

### Weil's Riemann Hypothesis for dummies?

Weil's Riemann Hypothesis is a deep result that I don't fully understand, but it has understandable corollaries which interest me. For example:
(a) For any projective curve $X$ satisfying certain ...

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**0**answers

301 views

### Conjectures on fractions where each digit appears once in numerator and denominator

This is a highly redacted version of a question that was asked before. Please see Criteria of considering relevance of the question to the domain of research topics for details.
Some numerical ...

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151 views

### Kloosterman-like sum with inverse to different moduli

In some recent work, the following strange-looking exponential sum arose:
$$
\sum_k \sum_r \sum_s e\bigg( \frac{r \bar s^{(r)} \bar k^{(r^2+s^2)}}{r^2+s^2} \bigg).
$$
Here $e(x) = e^{2\pi i x}$ as ...

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**0**answers

96 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) ...

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62 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 ...

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**1**answer

230 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 ...

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**2**answers

469 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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81 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 $S_0=P_{k\cdot b/2}(P_{b/2}(4))$ , ...

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**1**answer

380 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 ...

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86 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
...

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votes

**3**answers

608 views

### Number of primitive $n$th roots with positive versus negative real parts

Does anyone know a reference to the following results, which I can prove, but I suspect may be known. Let $R(n)$ denote the number of primitive $n$th roots of unity with positive real part, and $L(n)$ ...

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**0**answers

158 views

### In which orders can the numbers of prime factors of consecutive integers be?

Let $\omega(m)$ be the number of distinct prime divisors of a positive integer $m>1$. I am interested in the relative orders in which the numbers $\omega(n+1),...,\omega(n+k)$ can occur.
Given ...

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**5**answers

1k views

### Brief Introduction to Modular Forms

What are the best introductory texts on modular forms that are suited for a brief six week course intended for advanced undergraduates? The students will be quite sharp and as far as prerequisites go, ...

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208 views

### Where can I find the article of A. Borel: “Values of zeta-functions at integers, cohomology and polylogarithms”? [closed]

Where on the internet can I find this article?
I know that it is in this book: Current trends in mathematics and physics, Narosa, New Delhi, 1995.

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votes

**1**answer

339 views

### Is SOC known to imply the Grand Riemann Hypothesis? [closed]

I'm currently working on a conditional proof of the Grand Riemann Hypothesis, which is based on the assumption that every field automorphism of $\mathbb{C}$ that commutes with an element of the ...

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231 views

### Conjectured congruence for the Apery numbers

Numerical evidence for the first hundred Apery numbers
$$A_n=\sum\limits_{k=0}^n\binom{n}{k}^2\binom{n+k}{k}^2$$
suggests the following congruence relation
$$A_n\equiv 0\; (\mathrm{mod}\; ...

**2**

votes

**1**answer

271 views

### Is there an english translation of Delignes “La conjecture de Weil pour les surfaces K3.”?

The title is pretty self explanatory: I'm looking for an english translation of Delignes inventiones paper "La conjecture de Weil pour les surfaces K3."
Anyone know if such a thing exists?
Thanks!

**1**

vote

**1**answer

139 views

### Link between integral points on varieties and solutions to Diophantine equations

Let $k$ be a number field, $S$ a finite set of places of $k$ including the infinite ones and $F(X_1,\dots,X_n)$ a polynomial in $k[X_1,\dots,X_n]$.
I am looking for notes, books or surveys detailing ...

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vote

**0**answers

153 views

### Reference for Local class field theory via witt vectors

I would like to find some books or lecture notes on geometric local class field theory via Witt vectors. I can't find any good paper on this subject.All approaches in the books to local class field ...

**2**

votes

**0**answers

122 views

### n-ary quadratic forms with $S$-integer values

Let $Q(x_1,\ldots,x_n):=x_1^2+\cdots+x_n^2$ be an $n$-ary quadratic form.
Given a finite set of (rational) primes $S$ is there an algorithm or theorem that describes all solutions to ...

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votes

**0**answers

195 views

### Irreducibility of Galois representations attached to unitary groups

If $G$ is a unitary group in $n$ variables over $\mathbb Q$, attached to an hermitian form for an imaginary quadratic extension $E/\mathbb Q$ and if we suppose that the hermitian form is definite over ...

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votes

**1**answer

708 views

### How strong is this conjecture? $(Z/nZ)^*$ is generated by “small” elements

Conjecture: There are constants $c,k$ such that every $(Z/nZ)^*$ is generated by its elements smaller than $k (\log n)^c$.
Where $(Z/nZ)^*$ is the multiplicative group of integers mod $n$. My ...

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votes

**1**answer

309 views

### Class groups of orders

In Cox's book "Primes of the form $x^2 + ny^2$", he proves that in a quadratic imaginary field $K$, if $\mathcal O$ is an order of conductor $f \in \mathbb Z$, we have that the class group ...

**1**

vote

**1**answer

111 views

### Quotients of number rings IZ[zeta_l]

Let $l=p^r$ a prime power and $\zeta$ a primitive l-th root of unity. It is classical result, that $(1-\zeta)^{\varphi(l)}=p\cdot\epsilon\in\mathbb{Z}[\zeta]$ for a unit $\epsilon$.
It should be a ...

**11**

votes

**2**answers

408 views

### References for particular topics related to Langlands

I have never really concentrated on Langlands, which explains my poor level of understanding of it. But I have read quite a few introductory papers related to Langlands, and to the circle of ideas ...

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votes

**2**answers

273 views

### Reference for $p$-adic Hodge theory with coefficients

Let $K$ be a $p$-adic field and $L$ be a finite or infinite extension (maybe algebraic ?) of $\mathbb{Q}_p$.
Is there a reference for $p$-Hodge theory for representations $\rho : Gal_K \rightarrow ...

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votes

**1**answer

534 views

### Continued fraction representation of Zeta

A question at math.SE is asking for references. The fraction is quite nice! Check it out and post some references if you know of any.
I found this at arxiv, but it doesn't apply to Zeta.

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vote

**2**answers

301 views

### How to prove $\mathop {\lim }\limits_{x \to \infty } \sum\limits_{{f_x}(p) = 1} {\frac{1}{p}} = \ln 2$ for $p \le x$?

Let ${f_x}(m) = \sum\limits_{\left. p \right|m} {{f_x}(p)}$ be a
strongly additive function on positive integer number $m$, where $p$ is a prime number. Set
$${f_x}(p) = \left\{ ...

**2**

votes

**1**answer

327 views

### Looking for paper: Weil's original 1952 “Sur les formules explicites de la théorie des nombres premiers”

I am looking for a source (preferably online) for Weil's original 1952 paper on the explicit formula. I am aware of an english translation available here, but would like to have access to the original ...

**4**

votes

**0**answers

131 views

### Is there a version of Serre's modularity conjecture for projective representations?

Serre's modularity conjecture asserts that a continuous odd irreducible representation $$\overline{\rho} : G_\mathbb{Q} \rightarrow \mathrm{GL}_2(\mathbb{F}_q)$$ must be modular, in the sense that ...

**2**

votes

**2**answers

166 views

### Looking for a reference to a classical formula for the sum of the base-$b$ digits of an integer

Pick an integer $b \ge 2$, and for $n \in \mathbf N$ let $s_b(n)$ denote the sum of the base-$b$ digits of $n$. It is a nice exercise to prove that $$s_b(n) = (b-1) \sum_{i=1}^\infty ...

**22**

votes

**2**answers

1k views

### Was Vinogradov's 1937 proof of the three-prime theorem effective?

Was Vinogradov's first proof of the three-prime theorem effective?
Reasons for my question: Vinogradov presented his proof in 1937 in a monograph; the English translation by K.F. Roth and A. ...

**3**

votes

**2**answers

193 views

### equivalence of quadratic forms over finitely generated fields

Over number fields, two quadratic forms are equivalent iff they have the same dimension, signature, discriminant and Hasse invariant.
How is the situation like over finitely generated fields?

**4**

votes

**2**answers

206 views

### Asymptotics of special square-free numbers

What is the asymptotic number of square-free numbers less than $x$ with exactly $k$ prime divisors?