**1**

vote

**2**answers

269 views

### Exponential Sum Bound

In
http://131.220.77.52/files/preprints/diophantine/bruedern/wpminicubefinal.pdf, Bruedern and Wooley mention the following fact on the bottom of page 6:
Let
...

**2**

votes

**2**answers

133 views

### Reference for Skinner-Urban on the Iwasawa main conjecture for $GL_2$

Does anyone know the existence of an expository paper or a report discussing the work of Skinner-Urban
"The Iwasawa main conjecture for $GL_2$"?
I am interested in partucular in the case of elliptic ...

**1**

vote

**1**answer

113 views

### Quadratic Gauss sums: Explicit determinations?

Can anyone please tell me (give me a reference, preferably) if there is any explicit determination of sums of the form $g(n,\chi):=\sum_{r=1}^{q}\chi(r)e(\frac{rn+r^2}{q})$ where $\chi$ is a Dirichlet ...

**2**

votes

**1**answer

120 views

### Existence of real modular function with specific behavior as $q\to 0$

I am looking for a real modular function $F(q,\bar{q})$ such that in the limit of small $q,\bar{q}$ it behaves as:
$F(q,\bar{q})=(a_0 + a_1 (q + \bar q)+...)\log q \bar q+ (b_0 + b_1 (q + \bar ...

**5**

votes

**3**answers

331 views

### Bounds on horizontal minima of the Riemann zeta function

It is known that $\zeta(s)$ has an infinity of zeros in the strip $0<\sigma<1$ and that those zeros become closer together as $t\rightarrow\infty$. More precisely, Littlewood showed that there ...

**2**

votes

**1**answer

236 views

### Does the set of automorphisms of a cyclic group exhibit some sense of randomness?

I prefer to proceed with a concrete example if I may. I appreciate that the answer might well be better explained with group theory, geometry and/or notions from probability theory, which I welcome.
...

**2**

votes

**0**answers

41 views

### On the uncountability of a subset of U-numbers of type $\leq m$

We say that $\xi\in \mathbb{R}$ is an $m$-ultra number if there exists a sequence $(\alpha_n)_n$ of $m$-degree real algebraic numbers, such that
$$
|\xi-\alpha_n|<(\exp^{[3]}(H(\alpha_n)))^{-n},\ ...

**1**

vote

**1**answer

178 views

### Is it possible to sum the divergent series with prime coefficients?

It is known that the series $$ P := \sum_{n=1}^{\infty} p_{n} \qquad \text{where } p_{n} \text{ is the n'th prime} $$ cannot be summed by means of (prime) zeta function regularization. (The result was ...

**0**

votes

**1**answer

183 views

### A Problem Concerning Odd Perfect Number

Briefly, prove that every odd number having only three distinct prime factors cannot be a perfect number.
I know there are results much stronger than the one above, but I am looking for an answer ...

**9**

votes

**3**answers

943 views

### Finding integer points on elliptic curves via divisibility conditions like $(a+b)^2 \mid (2b^3+6ab^2-1)$

Is the following conjecture correct?
Conjecture. The divisibility condition $(\alpha+\beta)^2 \mid (2\beta^3+6\alpha\beta^2-1)$ has no solutions in positive integers $1 \le \beta < \alpha < ...

**4**

votes

**1**answer

132 views

### Constructing a family of convergents from continued fractions formed by a set of prime partial quotients

For a given real number $x$, the continued fraction representation $x = [a_0; a_1, a_2, \cdots]$ where $(a_n)_{n \geq 0}$is defined by setting $x = \alpha_0$, then $a_i = \lfloor \alpha_i \rfloor$, ...

**7**

votes

**1**answer

193 views

### Is the sequence of Apéry numbers a Stieltjes moment sequence?

Consider the sequence of Apéry numbers
$$
A_n = \sum_{k=0}^n \binom{n}{k}\binom{n+k}{k}\sum_{j=0}^k \binom{k}{j}^3
= \sum_{k=0}^n \binom{n}{k}^2\binom{n+k}{k}^2 .
$$
In an email, physicist Alan Sokal ...

**2**

votes

**0**answers

69 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}= ...

**2**

votes

**1**answer

175 views

### Dynamics in the integers - Floor function

Let $\alpha$ be an irrational with $0<\alpha<1$. Consider the function given by \begin{align*}
f: &\mathbb{N}\longrightarrow \mathbb{N}\\ &x\longmapsto [ \alpha\cdot x]\end{align*} where ...

**0**

votes

**0**answers

189 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 ...

**1**

vote

**2**answers

220 views

### Rationale behind an requirement on Turing machines

Hopcroft and Ullman's definition of a Turing machine seems to be standard. This definition defines a Turing machine to be a 7-tupel $T = \langle Q,\Gamma,b,\Sigma,\delta,q_0,F \rangle$ obeying some ...

**-1**

votes

**0**answers

105 views

### Primality matrices [on hold]

This question is some kind of a follow-up to my previous thread untitled About Goldbach's conjecture, the content of which follows: 'let's consider a composite natural number $n$ greater or equal ...

**6**

votes

**1**answer

210 views

### Rational points techniques on curves not using their Jacobian

Let $C/K$ be a curve of genus > 2 over a number field $K$ and suppose there exists a $p \in C(K)$. Then a recurring theme in studying $C(K)$ is using the map $C \to J(C)$ normalized by sending $p$ to ...

**8**

votes

**1**answer

751 views

### What is the arithmetic Nullstellensatz?

The only precise statement (coming from a reliable source) of the "arithmetic Nullstellensatz" I can find is in Gowers's book, stating that two polynomials with integral coefficients have the same ...

**0**

votes

**0**answers

61 views

### Must the radical of polynomial evaluated at integers be small enough at least once?

Basically I am interested if the radical of polynomial evaluated at integers can be small enough at least once.
Let $f \in \mathbb{Z}[x], \deg(f)>1$ be squarefree. For integer $a$ and $f(a) \ne 0$ ...

**0**

votes

**1**answer

325 views

### Counterexample to Pólya's conjecture

It is known that Polya's conjecture is false and the smallest counter-example is about $10^9$.
Assuming that we are searching for a counter-example not knowing that it exists. What useful information ...

**0**

votes

**1**answer

135 views

### Composition of a transcendental function with a rational function [on hold]

The problem is: let $f: \mathbb{R}\to \mathbb{R}$ be an analytic transcendental function and let $\psi(x)=\frac{x}{2(1+x^2)}$. Is the function $f(\psi(x))$ transcendental?

**46**

votes

**5**answers

3k views

### Strange (or stupid) arithmetic derivation

Let us consider the following operation on positive integers: $$n=\prod_{i=1}^{k}p_i^{\alpha_i} \qquad f(n):= \prod_{i=1}^{k}\alpha_ip_i^{\alpha_i-1}$$ (Is it true that if we apply this operation to ...

**1**

vote

**2**answers

229 views

### Asymptotics on prime divisors

Somewhat inspired from the Zsigmondy's theorem is my question. Suppose Let $a_{1}> a_{2}> \cdots > a_{k}$ be nonzero integers, with $k \geq 2$. Let $a(n) : = a_{1}^{n}+\cdots+a_{k}^{n}$ for ...

**8**

votes

**1**answer

263 views

### Separation of lattice points on the Mordell elliptic curve

Consider the Mordell equation x^3 – y^2 = k, where x is a non-square positive integer and y^2 is the perfect square nearest to x^3. Noam Elkies (see http://www.math.harvard.edu/~elkies/hall.html) ...

**9**

votes

**1**answer

222 views

### Repetend digit graphs for $1/n$ in base $b$

Here is a decimal expansion of $\frac{1}{34}$:
$$(1/34)_{10}=0.02941176470588235\overline{2941176470588235}\ldots$$
And here is a graphical representation of the 16-digit
"repetend," as a directed ...

**6**

votes

**4**answers

725 views

### Proofs of the Chevalley-Warning Theorem

A well known proof of the Chevally-Warning Theorem is the one listed on wikipedia: http://en.wikipedia.org/wiki/Chevalley%E2%80%93Warning_theorem
Are there any other proofs of this, or ...

**2**

votes

**0**answers

119 views

### Metric on the set of subsets of the rational primes

I was thinking how to say that two sets of prime numbers are close to each other, and came up with this.
NOTATION
$\ \Delta(A\ B)\ :=\ (A\setminus B)\cup(B\setminus A)\ \ $ is the symmetric ...

**2**

votes

**1**answer

191 views

### A family Mersenne composite numbers?

I believe that the number
$$2^{2^{2t+1}+2t-1}-1$$
is composite for all positive integer $t$. I tested this for many $t$'s, but so far I didn't get a proof. Any idea?

**4**

votes

**1**answer

133 views

### Estimates on derivatives of Bessel function

In Duke, Friedlander and Iwaniec's Erratum on "Bounds for automorphic L–functions. II"
They have the following estimates for derivatives of Bessel functions: For $k \geq 2$
\begin{align}
& ...

**-1**

votes

**0**answers

167 views

### Where is the Flaw in the Argument? [on hold]

Some days ago I have read about the Legendre Symbol. I found that there is no easy method for computation of the symbol. I tried to find out a method for determining the value of ...

**1**

vote

**0**answers

74 views

### On the computation of Asai L-function

I want so compute some simple twisted Asai L-function.
Let $E/F$ be a quadratic extemsion of number fields and $v$ a finite place of $F$.
Let $\chi$ be a unitary automorphic character of ...

**2**

votes

**1**answer

474 views

### Efficient algorithms to determine the roots of: $p(x) = r^x $ in the finite field $GF(q)$, where $r$ a primitive root of the field

I need to make sure that no efficient (i.e., polynomial time) algorithm exists for the following problem:
Exponentiating Polynomial Root Problem (EPRP)
Let $p(x)$ be a polynomial with $\deg(p) \geq ...

**6**

votes

**2**answers

255 views

### Smooth sums of coprime smooth integers

Observe that for any $\epsilon > 0$ there are infinitely many triples of
$c^\epsilon$-smooth coprime positive integers $a$, $b$ and $c$ such
that $a + b = c$. -- Considering triples of the form ...

**0**

votes

**0**answers

171 views

### Which of the Mochizuki's works are the most closely related to elliptic curves?

I'm very much interest about algebraic geometry and number theory along with cryptography, but I have a special interest about the elliptic curves. I have heard a lot of interesting things about ...

**-1**

votes

**0**answers

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 ...

**0**

votes

**0**answers

79 views

### Arthur's refinement of parameters for unitary automorphic representations

In his work on the classification of automorphic representations of a group $G$, Arthur has conjectured that the parameterization of such representations involves a homomorphism $\rho : SL_2 \times ...

**1**

vote

**1**answer

143 views

### For a defined set $M$ (see problem) do there exist $a,b$ natural numbers so that $a,ab+1 \in M$

Let $\rho \in \mathbb{R}\setminus \mathbb{Q}$ be a irrational nuber, and
let $\varepsilon>0$ be arbitrarily small. Define $M=\{m \in \mathbb{N}: \exists k \in \mathbb{N}\hbox{ so that} |\rho m -k ...

**0**

votes

**0**answers

86 views

### A System of Diophantine Equations [closed]

$p^2+1=2y^2$
$p+1=2x^2$
$p$ is prime and $x,y$ are integers. I conjecture that this only has solution for $p=7$

**2**

votes

**0**answers

38 views

### Extension of the product formula for valuations to a simultaneous completion

It is well known that $\mathbb{C}$ and $\mathbb{C}_p$ are "algebraically" isomorphic (that is, ignoring the topology), but an isomorphism depends on the axiom of choice and there is no canonical way ...

**17**

votes

**1**answer

777 views

### If $\left(1^a+2^a+\cdots+n^{a}\right)^b=1^c+2^c+\cdots+n^c$ for some $n$, then $(a,b,c)=(1,2,3)$?

Question : Is the following conjecture true?
Conjecture : Let $a,b(\ge 2),c,n(\ge 2)$ be natural numbers. If $$\left(\sum_{k=1}^nk^a\right)^b=\sum_{k=1}^nk^c\ \ \ \ \ \cdots(\star)$$
for some $n$, ...

**4**

votes

**0**answers

105 views

### Without Skolem–Mahler–Lech Theorem? [closed]

Using Skolem–Mahler–Lech theorem one can easily prove the $\displaystyle \lim_{n\to +\infty}\left|\Re\left(\frac{1+i\sqrt{7}}{2} \right)^n\right| =+\infty$.
Is there a "simple way" to prove this ...

**7**

votes

**5**answers

582 views

### Small values of a polynomial evaluated at roots of unity

The MO answer http://mathoverflow.net/a/98176/11926 notes the following: Let $\gamma$ be an algebraic number that is not a root of unity. Then Baker's theorem implies that there is a constant ...

**10**

votes

**0**answers

369 views

### divisibility of Tamagawa numbers

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$. Let $p\ge11$ be a prime of good ordinary reduction for $E$ and assume that $p$ does not divide the degree of a minimal modular parametrization ...

**1**

vote

**2**answers

304 views

### Practical use of estimates for the Gauss Circle Problem

This question is related to this and this ones. The Gauss Circle problem asks for the number $N(r)$ of integer points within a sphere of radius $r$ centered at the origin. It is well known that $N(r) ...

**1**

vote

**1**answer

92 views

### On the pole of local L-function

Let $F$ be a number field and $v$ a finite place of $F$.
Let $\chi_v$ be a unramified unitary character of $F_v$.
Then we define local L-function $L_v(s,\chi_v):=\frac{1}{1-\chi_v(\omega)q^{-s}}$ ...

**4**

votes

**0**answers

158 views

### Non-hyperelliptic families of curves with trivial Ceresa class (or Gross-Schoen class)

Suppose X/K is a curve over a field K, which we want to think of as non-algebraically closed, and let x be a point of X(K). The Ceresa cycle is defined as follows; you can embed X in Jac(X) by sending ...

**10**

votes

**5**answers

2k views

### Rational points on a sphere in $\mathbb{R}^d$

Call a point of $\mathbb{R}^d$ rational if all its $d$ coordinates are rational numbers.
Q1.
Is the unit sphere $S :\; x_1^2 +\cdots+ x_d^2 = 1$ dense in rational points, i.e. does $S$ include a ...

**0**

votes

**0**answers

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 ...

**3**

votes

**1**answer

515 views

### Possibly an easy application of Tate-Nakayama duality and local CFT

Let $F$ be a non-archimedian local field and $F^{ur}$ the maximal unramified extension. Let $T$ be a torus defined over $F$ such that $T$ splits over a tamely ramified extension and is anisotropic ...