**0**

votes

**1**answer

40 views

### Equivalence between Diffie Hellman and Discrete Log

For which non-trivial groups, do we know that the Diffie Hellman problem and the Discrete Log are equivalent?
Is there any group for which we suspect them to be different?
Could there be a finite ...

**7**

votes

**1**answer

136 views

### “Pythagoras number” for integral matrices

It is classically known that every positive integer is a sum of at most four squares of integers, i.e. every sum of squares of integers is a sum of four squares of integers. Now consider a symmetric ...

**6**

votes

**1**answer

244 views

### confounding riddle about fine moduli schemes and twists of elliptic curves

I've encountered a strange situation while thinking about modular curves... Consider the modular curve $Y(3)$ parametrizing elliptic curves with a symplectic basis for their 3-torsion. This curve has ...

**2**

votes

**0**answers

160 views

### Avoiding Chinese Remainder Theorem

Given $k\in\Bbb N$ with $k<(\log_2N)^{\frac1\alpha}$ where $\alpha>2$ is fixed and $N$ being some integer such that $$N<\prod_{i=1}^k\pi_i^{a_i}$$ where $\pi_1,\pi_2,\dots,\pi_{k-1},\pi_k$ ...

**10**

votes

**2**answers

288 views

### A condition for a sequence defined by a recurrence relationship to all be integers

I am interested in a specific sequence $\{a_n \}$ defined by a simple recurrence relationship: $$a_n = \frac {a_{n-1} ^2 +c} {b} $$ where $b,c\in \mathbb{Z}$. I want to find all $b,c$ such that there ...

**6**

votes

**2**answers

315 views

### Frobenius elements in infinite extensions

Let $K$ be a number field, $\bar K$ an algebraic closure and $G$ the associated absolute Galois group. How can I define the Frobenius elements of $G$ or at least their conjugacy class?
I know how ...

**2**

votes

**1**answer

100 views

### Ramification of prime ideal in Kummer extension

Let $\mu \in \mathbb{Q}(\zeta_n)$ lie above the rational prime $p$, and let the prime ideal $\mathscr{P}\subset \mathbb{Z}[\zeta_n]$ have ramification index $a$ over $\mu$.
Why is it then true that ...

**0**

votes

**0**answers

82 views

### Bounding $\dfrac{r(x)}{\pi(x+r(x))-\pi(x-r(x))}$ with $1\ll r(x)\ll \log^{4}(x)$

I would like to know whether it is possible to obtain the bounds $\sqrt{r(x)}\ll k(x)\ll r(x)$ where $k(x)={\pi(x+r(x))-\pi(x-r(x))}$ and $1\ll r(x)\ll \log^{4}(x)$ and thus ...

**5**

votes

**0**answers

113 views

### Zeros of eigenforms at a given elliptic curve

Let $N$ be an integer and $s \in X_1(N)(\mathbb C) = \Gamma_1(N) \backslash \mathbb H^*$, then one can define $T(s,N)$ to be the number of eigenforms in $S_2(\Gamma_1(N))$ that have a zero at $s$ [1].
...

**12**

votes

**0**answers

181 views

### Tiling a square with rectangles

Is it possible to completely tile a square with different rectangles of integer sides but all with the same area?
The original problem, not requiring integer sides for rectangles, was proposed by Joe ...

**2**

votes

**2**answers

298 views

### Primes $p$ for which $2p-1$ is prime

It's a well-known open problem (Sophie-Germain primes) whether there are infinitely many primes $p$, $2p+1$. What about $p$, $2p-1$?
Seemingly it's also an open problem (see here and the linked ...

**3**

votes

**1**answer

156 views

### Is $2^n -1$ finitely many times the product of consecutive primes? [duplicate]

This question was asked at MSE but recieved no attention at all.
Here it is:
Are there finitely many $(n,k) \in \mathbb{N}^2$ with $2^n-1=p_1p_2\cdots p_k$ ?
$p_1=3,p_2=5 , ...,p_k$ are ...

**4**

votes

**1**answer

155 views

### Another question on Heath-Brown's “Prime twins and Siegel zeros”

With a graduate student, I'm going through the paper (Proc. London Math. Soc. (3) 47 (1983), no. 2, 193–224.)
Here's the background and notation.
We have a quadratic character $\chi$ modulo $q$, ...

**5**

votes

**1**answer

106 views

### History of spectral methods to the study of real analytic $GL_2$-Eisenstein series

I'm trying to sort out the history of spectral methods in the study of real analytic $GL_2$-Eisenstein series. From what I read so far, I would say that the subject was really kicked off by the ...

**-1**

votes

**0**answers

111 views

### How does this small change in the Pollard Rho method affect its complexity? [on hold]

In finding the smaller factor $p$ of an input number $n$, the Pollard Rho method takes time bounded by a function in $O(\sqrt{p})$. (Did I get that right?)
Now let's say I tweak the method just a ...

**3**

votes

**1**answer

152 views

### Examples when one can use the the symmetric power $L$-functions to study topics related to the number theory

"The symmetric power $L$-functions are a powerful tool for studying algebraic or geometric objects through analytic methods." I read this sentence in the introduction of a Master thesis. I want to ...

**6**

votes

**1**answer

181 views

### Simplest case of Langlands-Shahidi method

I would like to read the simplest examples of Langlands-Shahidi method carried out to prove the functional equation of $L$-function.
Could the constant term of $\mathrm{GL}(2)$-Eisenstein series be ...

**7**

votes

**2**answers

247 views

### Average digit sum in different bases

Given two natural numbers $n\geq 1$ and $b\geq 2$, denote by $S_b(n)$ the sum of the digit of $n$ in its representation in base $b$. Clearly $S_b(n)$ varies from 1 (when $n$ is a power of $b$) to ...

**3**

votes

**0**answers

96 views

### The density of square-free integers represented by a cubic polynomial

Suppose that $f(x)$ is an irreducible cubic polynomial with integral coefficients. Suppose further that for all primes $p$, there exists an integer $n_p$ for which $p^2 \nmid f(n_p)$. Then it is a ...

**7**

votes

**1**answer

129 views

### Structure of Deligne-Lusztig representations $R_{T,\theta}$ for ministropic $T$ and cuspidal representations

Let $G$ be a reductive group over a finite field $k$, let $F$ be a Frobenius morphism on $G$.
I'll start with a somewhat vague question and make my question more specific further down:
How do ...

**3**

votes

**1**answer

94 views

### Bibliography suggestion for Kummer theory

I already posted a question about a sum involving the degree of a Kummer extension.
Now I'm interested in a more specific fact about Kummer extensions.
From Hooley's paper "On Artin's conjecture", we ...

**-1**

votes

**0**answers

33 views

### Prime dividing binomial coefficient involving prime power [migrated]

I was wondering if there was a straightforward proof of the following fact (which I can show is true for specific cases, but not generally):
Let $n$ be composite, and let $p$ be a prime factor $n$. ...

**0**

votes

**0**answers

58 views

### Does every odd prime divide a Mersenne number? [migrated]

For a given odd prime number $p$, is there an integer $n>0$ such that $p$ divides $2^n-1$?

**2**

votes

**1**answer

117 views

### Doubt concerning a sum involving Kummer extension degrees

I'd like to estimate the following sum
$$
\sum_{n\leq x}\frac1{k_n}\;,\qquad x\rightarrow \infty\;,
$$
where
$k_n=[\mathbb{Q}(\zeta_n,a^{1/n}):\mathbb{Q}]$
is the degree of a Kummer extension for a ...

**2**

votes

**1**answer

83 views

### Parallel algorithm for modular multiplication of polynomials over Z/nZ

Is there a parallel algorithm for doing modular multiplication of polynomials over Z/nZ? n is a very large number (for hundreds and thousands of bits).
Normally, the method used is binary ...

**-6**

votes

**0**answers

166 views

### Power equivalence of two positive real numbers [closed]

I posted the same question on Math.Stackexchange but I didn't get any precise answer until now. Thus I asked it here.
Assume $a,b>0$ are two real numbers. Define the sequences $a_n, b_n$ as ...

**5**

votes

**3**answers

244 views

### Smooth complete intersections and sharpness of the Chevalley-Warning theorem

Let $X$ be a complete intersection in $\mathbb{P}^n$ of multidegree $(d_1,\ldots,d_r)$. If we're working over a finite field $\mathbb{F}_q$, the Ax-Chevalley-Warning theorem says that if $X$ is in the ...

**2**

votes

**0**answers

87 views

### Additive combinatorics and a Diophantine equation

Let $(n_k)_{1 \leq k \leq N}$ be a sequence of distinct positive integers. For $v \in \mathbb{Z}$ set
$$
A_N(v) = \# \Big\{ (k,\ell) \in \{1, \dots, N\}^2, ~k \neq \ell:\quad n_k - n_\ell = v \Big\}.
...

**0**

votes

**0**answers

59 views

### Improvement on $\phi\sigma$ bound [migrated]

We have:
$$\dfrac{6}{\pi^2}\lt\dfrac{\phi(n)\sigma(n)}{n^2}\le1$$
with equality iff $n=1$.
Wikipedia
Are there any known improvements on these bounds?

**2**

votes

**1**answer

141 views

### Kloosterman sum

Does anybody know the non-trivial bound for this sum?
$S(m,n,c,q)=\sum_{a,b\in \mathbb{Z}/cq\mathbb{Z}, \;ad\equiv 1\text{ mod }c} e^{2\pi i(am+nd)/qc},$
where $m,n\in\mathbb{Z},\;q,c\in\mathbb{N}$.
...

**2**

votes

**0**answers

59 views

### Determining coefficients of a Dirichlet series based on values on a vertical line

Let us suppose we have a Dirichlet series
$$ D(s) = \sum_{n \geq 1} \frac{a(n)}{n^s},$$
and that we know the values of $D(\tfrac{1}{2} + im)$ for $m \in \mathbb{Z}$. Can we recover the coefficients ...

**4**

votes

**0**answers

84 views

### Behavior of the “mean prime factor” of numbers

This question concerns the behavior of
a function $f(\;)$ that maps each number in $\mathbb{N}$ to
its mean prime factor.
I previously posted premature questions, now deleted, which
explains the cites ...

**27**

votes

**3**answers

726 views

+300

### Simple argument regarding sums of two units in a number field?

I wonder if it is possible to show, without using the Schmidt subspace/Roth theorem/Baker's bounds on linear forms in logarithms or other very deep results, that, in a number field, not all integral ...

**0**

votes

**0**answers

47 views

### Factorial Sums over Compositions or ``Unlabeled Permutations"

Let $C_n$ denote subset of integer compositions of $n$ and $c=(c_1,c_2,\dots c_n)$
In a divergent sum, the sequence
$$
a_n=\sum_{c\in C_n} \prod_{c_i\in c} c_i!
$$
frequently shows up and one ...

**-7**

votes

**0**answers

146 views

### Formal generic fibre and Fermat's Last Theorem

Set $A_{n} \colon= {\Bbb F}_p[[S_1,...,S_n]]$ and
$A_{n,d} \colon= {\Bbb F}_p[[S_1,...,S_n]][[X_1,...,X_d]]$ be a $d$-variables formal power series ring over $A_n$. We denote by $K$ the fractional ...

**8**

votes

**1**answer

291 views

### Distribution of the number of prime factors

Count the number of prime factors of a number $n$
to include multiplicity,
so that
$$n=24=2^3 \cdot 3 = 2 \cdot 2 \cdot 2 \cdot 3$$
has $4$ prime factors, and
$$n =
6500 =
2^2 \cdot 5^3 \cdot 13 =
2 ...

**11**

votes

**1**answer

232 views

### Generating function of the Thue-Morse sequence

Let $T$ be the generating function of the Thue-Morse sequence; thus,
$T(x)=x+x^2+x^4+x^7+\dotsb$. It is known that $T$ satisfies the nice
congruence
$$ (1+x)^3 T^2(x) + (1+x)^2 T(x) + x \equiv 0 ...

**6**

votes

**0**answers

220 views

### Is Frac $\mathbb{Z}((x))$ Hilbertian?

Note that Frac $\mathbb{Z}((x)) \ne\mathbb{Q}((x))$.
As a result of Some questions about the ring Z((x)), we know that it is a Dedekind domain with uncountably many primes, each of which is of the ...

**3**

votes

**1**answer

238 views

### Did Erdős prove there are two primes $4a+1, 4b+3$ between between $n$ and $2n$?

http://mathworld.wolfram.com/ChoquetTheory.html
Is the claim in the link true? Here's the reference given there:
https://www.renyi.hu/~p_erdos/1934-01.pdf
Erdős proved that there exist at least ...

**8**

votes

**1**answer

350 views

### Do the complex zeros of the sum/difference of these series all reside on the line $\Re(s)=\frac12$?

The following series seems convergent for all $s\in \mathbb{C}$:
$$\displaystyle f(s):=\sum_{n=1}^\infty \frac{(-1)^n}{(n+s)^{n+s}}$$
The function itself does not appear to have any real or complex ...

**22**

votes

**2**answers

450 views

### Are there irreducible polynomials with all zeros on two concentric circles?

This is somewhat similar to this recent question, but extending in a different direction.
Let $f(x)$ be an irreducible polynomial of degree $n$ with integer coefficients. Call such $f$ a bicycle ...

**6**

votes

**1**answer

143 views

### Young-Fibonacci lattice and purely periodic continued fractions

The Fibonacci lattice $\mathcal{F}$ is the poset of all finite words consisting of 1's and 2's where a word $v$ covers a word $u$ if $v$ is obtained from $v$ by either (a) inserting a 1 in $u$ prior ...

**13**

votes

**1**answer

422 views

### Erdös-Turán via Hardy-Littlewood circle method?

For any set $B\subseteq \mathbb{N}$ one can associate the formal series
$$f_B(z) = \sum_{b\in B}z^b$$
and obtain
$$f_B(z)^k = \sum_{n\geqslant 0} r_{B,k}(n)z^n,$$
where $r_{B,k}(n) = ...

**5**

votes

**2**answers

286 views

### Inverse Galois problem for simple Lie type groups

Progress towards the Inverse Galois problem over $\mathbb{Q}$ is very well documented for sporadic groups ($M_{23}$ is the only case open) and for $PSL_n(q)$ (a lot of cases known, but wide open in ...

**5**

votes

**1**answer

438 views

### Intuition behind salient numbers in number of h-cobordism classes of smooth homotopy n-spheres

The Wikipedia article on Exotic Sphere displays this sequence of numbers (see also OEIS A001676 and the Milnor link therein) for the order of the classses as
$$1, \;1, \;1,\; 1,\; 1, \;1, \;28,\; ...

**4**

votes

**1**answer

243 views

### Listing all solutions to $n = x^2 + y^2 + z^2 $ with integers

I would like to list all ways of writing $n$ as the sum of 3 squares. This is slightly different from finding just one:
Is there an algorithm for writing a number as a sum of three squares?
...

**2**

votes

**1**answer

329 views

### Does Borel's proof for existence of normal numbers make an essential use of axiom of choice?

A normal number is a real number whose infinite sequence of digits in every base $b$ is distributed uniformly in the sense that each of the $b$ digit values has the same natural density $\frac{1}{b}$, ...

**9**

votes

**1**answer

178 views

### On Bailey–Borwein–Plouffe formula for irrational numbers

A BBP-type formula for an irrational number $\alpha$ in the integer base $b\geq 2$ is a formula in the form $\alpha=\Sigma_{k=0}^{\infty}\frac{1}{b^k}\frac{p(k)}{q(k)}$ ($p, q$ are polynomials in ...

**10**

votes

**1**answer

184 views

### The density of integers represented by a binary form

Suppose that $F(x,y)$ is a binary form of degree $d \geq 3$ with integral coefficients, and non-zero discriminant. It is known (from a paper due to Erdős and Mahler from 1938) that the density of ...

**1**

vote

**0**answers

104 views

### A Question on Chinese Remainder Theorem [closed]

Let $p_1,p_2,\ldots,p_n$ be odd primes.
Let $y$ be the unique solution of the Chinese Remainder Problem ( $0 \le y < m$, $ m = p_1\cdot p_2 \cdots p_n$)
$ y = \frac{p_1-1}{2} \text{ mod } (p_1) ...