**0**

votes

**0**answers

16 views

### Function series involving a suite of imprimitive Dirichlet characters and a zero of $L(\chi,s)$

I have difficulties to understand the behavior of following suite of functions near zero :
$$F_P(x)= \sum\limits_{n=P}^{\infty} \chi_P(n) f(nx)$$
With $f(x) = x^{-s_0} e^{-x}$ where $s_0$ is a non ...

**2**

votes

**0**answers

67 views

### Differences associated with differences of primes: are they all 1,2,3?

Let $d_k$ be the $k^{th}$ difference sequence of the primes; that is, $$d_k = \sum_{i=0}^{k} (-1)^i {k \choose i} P_{k+1-i},$$ where $P_i$ denotes the $i$-th prime number. Let $(s_n)$ be the ...

**-1**

votes

**1**answer

52 views

### If $q^k n^2$ is an odd perfect number with Euler prime $q$, are the following statements known to hold in general?

Let $\sigma$ be the classical sum-of-divisors function.
A number is said to be perfect if $\sigma(N)=2N$.
If $q^k n^2$ is an odd perfect number with Euler prime $q$ (i.e., $q \equiv k \equiv 1 \pmod ...

**0**

votes

**0**answers

93 views

### A diophantine equation

A few days ago I saw a question about the diophantine equation $p^2+p+1=q^\alpha$ in
What is prime power of this equation of p?
and later in
A Diophantine equation with prime powers
I want the ...

**3**

votes

**0**answers

44 views

### Reference for a lemma on the asymptotic upper density of special sets with large gaps and intervals

Update. Based on Anthony Quas' comment below, the proof can be made sensibly shorter and the lemma can be slightly generalized by weakening the old assumption (iii).
In a joint paper that I am ...

**8**

votes

**2**answers

254 views

### On the number of consecutive divisors of an integer

Define for $n \in \mathbb{N}$ the function $$\tau_1(n):=\sum_{\substack{d|n, \\ d+1|n}}1,$$ i.e. the number of consecutive divisors of an integer. The average of $\tau_1(n)$ is $1$ since $$\sum_{n\leq ...

**8**

votes

**0**answers

404 views

### Algorithmic (un-)solvability of diophantine equations of given degree with given number of variables

Question: For which $d, k \in \mathbb{N}$ is there an algorithm to determine
whether a polynomial diophantine equation
$$
P(x_1, \dots, x_k) = 0, \ \ \ P \in \mathbb{Z}[x_1, \dots, x_k]
$$
...

**1**

vote

**0**answers

84 views

### Level-Lowering in Weight 2

Let $N$ and $p$ be relatively prime integers with $p$ a prime. Suppose $f$ is a weight $k=2$ (normalized, cuspidal, etc) newform of level $\Gamma_1(N) \cap \Gamma_0(p)$. I seem to recall the existence ...

**-3**

votes

**0**answers

163 views

### Could RH be a consequence of some kind of central limit theorem? [on hold]

In the last issue of "Pour la Science" (French edition of Scientific American), there is an article about random geometry on the sphere where the authors invoke the central limit theorem to explain ...

**-1**

votes

**0**answers

63 views

### how many pythagorean triplets can be formed with N given so that addition of all three sides is equal to N? [on hold]

Example:- N = 120
TRIPLETS = (30 ,40 ,50) , (20 ,48 ,52) , (24,45,51)

**-1**

votes

**0**answers

33 views

### Implementation of almost integer to cryptography [on hold]

Can there be any implementation of almost integers to create intractable problems relevant to public key cryptography?

**5**

votes

**0**answers

57 views

### divisibility by Bernoulli numbers of discriminant of Hecke algebra over the space of modular forms of level 1

For the space of modular forms and the space of cusp forms (here I only care about the level $1$ case), we have the action by Hecke algebras. Therefore, we can calculate the discriminant of this ...

**-5**

votes

**0**answers

162 views

### Are there “adelic” L-functions? [on hold]

Following Tom163's answer to this question, I would like to know whether L-functions defined through adelic representations (as defined in https://projecteuclid.org/euclid.em/1317758108) have been ...

**-4**

votes

**0**answers

43 views

### Periodicity of any fermat number modulo a prime [on hold]

It's simple to prove the recursive formula for Fermat numbers $F_n$ :
$F_{n+1} = ( F_n - 1 )^2 +1 $. From this , if one define the sequence $a_n = F_n \pmod p$ , where
$p$ is a odd prime , there's a ...

**-1**

votes

**0**answers

141 views

### A not-so-weak Goldbach's conjecture

While Goldbach's conjecture (every even integer greater than 2 can be expressed as the sum of two primes) remains open, one can weaken the question by asking whether every (even,odd) integer can be ...

**38**

votes

**4**answers

965 views

### How did Cole factor $2^{67}-1$ in 1903

I just heard a This American Life episode which recounted the famous anecdote about Frank Nelson Cole factoring $N:=2^{67}-1$ as $193707721 \times 761838257287$. There doesn't seem to be a historical ...

**4**

votes

**3**answers

303 views

### what is exactly the difference between the Selberg class and the set of Artin L-functions?

The question is in the title: from what I read in the answer to another question, Artin L-functions are conjecturally cuspidal automorphic L-functions for some algebraic group that can be transfered ...

**3**

votes

**2**answers

220 views

### For $k>3$ does there exist an odd prime $q_k$ such that $p_k=2^kq_k+1$ is prime and $p_k$ divides $a_k=\dfrac{3^{2^{k-1}}+1}{2}$?

Question: For $k>3$ does there exist an odd prime $q_k$ such that $p_k=2^kq_k+1$ is prime and $p_k$ divides $a_k=\dfrac{3^{2^{k-1}}+1}{2}$?\
If $k=3$ the answer is Yes because for $q_3=5$ we ...

**5**

votes

**1**answer

158 views

### Representing one diagonal of Pascal's triangle using special sums coming from a different diagonal

Let $m, n$ be any fixed natural numbers. Is it true that infinitely many elements of the sequence $\binom{m+k}{m}_{k=1,2,3,...}$ ( as well as of the sequence ...

**2**

votes

**1**answer

99 views

### References about identities of Gauss sum

I am reading the paper. In the end of page 10, there are the following identities of Gauss sum.
\begin{align}
& h(b) h(a+b) = q^b h(b) h(a), \\
& h(b) g(a+b) = q^b h(b) g(a), \\
& g(a+b) ...

**6**

votes

**0**answers

59 views

### Prime divisors of the norm of the first coefficient of an elliptic newform at width-1 cusps.

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ and let $f$ be its newform.
Suppose $p \geq 5$ is a prime such that $p^2 \mid N$. We assume $f$ is $p$-minimal, which is equivalent to that the ...

**6**

votes

**1**answer

206 views

### Are these inequalities for primes equivalent?

Let $p_n$ be the $n$th prime, let $L$ consist of the primes satisfying $p_{n+2} - 2p_{n+1} + p_{n} > 0$, and let $Q$ consist of the primes satisfying $p_{n+1}^2 < p_{n}p_{n+2}.$ Is $L=Q$?
...

**1**

vote

**0**answers

21 views

### Lower and upper density of iterations of subsets of $\mathbb{N}$ [migrated]

For $A\subseteq \mathbb{N}$ we define the lower and upper density by if $$\text{lowd}(A)=\text{lim inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}, \text{upd}(A)=\text{lim ...

**3**

votes

**3**answers

183 views

### Enumerating cosets of the modular group

Suppose we chose the generators $f(z) = z+1, g(z) = z-1, h(z) = -1/z$ for the modular group $\Gamma$ (i.e. the group of fractional linear tranformations $z \mapsto (az +b)/(cz+d)$ with $a,b,c,d \in ...

**2**

votes

**0**answers

159 views

### References for 'Theory of $p$-adic Galois Representations by Fontaine & Ouyang'

Presently I am reading the 'Theory of $p$-adic Galois Representations by Fontaine & Ouyang'. I am finding it difficult for eg. the initial sections on $l$-adic geometric representation of finite ...

**3**

votes

**0**answers

73 views

### Localized at $p$ integral representations of finite elementary $p$-groups

Let $C_p$ be a cyclic group of prime order $p$.
Let $F=C_p^n=C_p\times\dots\times C_p$ ($n$ times).
I would like to to classify finite dimensional representations of $F$ over ${\mathbb{Z}}$.
However, ...

**-1**

votes

**0**answers

65 views

### If $N = q^k n^2$ is an odd perfect number with Euler prime $q$, can $\sigma(n^2)$ be divisible by $(q+1)/2$?

If $N = q^k n^2$ is an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$), then can $\sigma(n^2)$ be divisible by $(q+1)/2$?
I think ...

**2**

votes

**1**answer

95 views

### Positive rational numbers as sum of unit fractions [duplicate]

Let $U = \{\frac{1}{n}: n\in\mathbb{N}, n>0\}$ be the set of unit fractions. For integers $m,n>0$ there is always a finite subset $S\subseteq U$ such that $\frac{m}{n} = \sum_{u\in S} u$, see ...

**7**

votes

**0**answers

114 views

### Complete list of exceptions and efficient algorithm for Waring's problem

2 weeks ago, Samir Siksek http://arxiv.org/abs/1505.00647 proved more than 150-years-old conjecture that every positive integer other than 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, 303, ...

**3**

votes

**1**answer

151 views

### Numbers represented by inhomogeneous forms

I have a family of Diophantine equations that I am trying to solve, and I am trying to figure out what methods could be used to prove existence of solutions. Unfortunately, the equations are ...

**2**

votes

**3**answers

552 views

### A Diophantine equation with prime powers

Let $p$ and $q$ be prime numbers such that $p^2+p+1=3q^a$: is it true that $a=1$?
This specific equation appears when computing order components of finite groups.

**3**

votes

**1**answer

310 views

### Is this version of van der Waerden's Theorem consistent with ZFC?

One way to phrase van der Waerden's Theorem is:
For every finite coloring of $\mathbb N$ and every finite $F \subseteq \mathbb N$, there exist $a,b \in \mathbb N$ such that $a + b \cdot F$ is ...

**10**

votes

**2**answers

503 views

### Does van der Waerden's Theorem hold for $\omega_1$?

One way to phrase van der Waerden's Theorem is:
For every finite coloring of $\mathbb N$ and every finite $F \subseteq \mathbb N$, there exist $a,b \in \mathbb N$ such that $a + b \cdot F$ is ...

**11**

votes

**2**answers

928 views

### What is prime power of this equation of p?

Let $p$ be a prime number, I think when $p^2+p+1=q^a$, where $q$ is a prime number, then $a=1$. But I can't prove it. Is it true?

**-1**

votes

**0**answers

100 views

### Order of element in algebraic group [migrated]

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...

**-5**

votes

**0**answers

58 views

### Better chances on a gambling game [closed]

I'm working on a program that I hope it will help me to get better chances on a gambling game. I have a very large database of numbers how occur in this game and my program pass through those numbers ...

**3**

votes

**0**answers

153 views

### The Prime Shuffle Puzzle

Given as input:
an ordered list of distinct prime numbers $A = ( p_1,p_2,...,p_n )$,
a set of triples $B = \{ (a_1, b_1, c_1 ), (a_1, b_1, c_1 ), ...., (a_m, b_m, c_m ) \}$, where $a_i, b_i, c_i ...

**3**

votes

**2**answers

342 views

### Non-standard Gauss sums

I have the following problem. Let $p$ be some prime. What is the value of
\begin{equation}
\sum_{k=1}^{p-1} \left(\frac{k+1}{p}\right) \omega_p^{kl},
\end{equation}
where $\left(\frac{k+1}{p}\right)$ ...

**6**

votes

**1**answer

130 views

### On the independence of lower and upper asymptotic and Banach densities

Given a set $X \subseteq \mathbf N^+$, denote by $\mathsf{d}_\ast(X)$ and $\mathsf{d}^\ast(X)$, respectively, the lower and upper asymptotic (or natural) density of $X$, viz. $$\mathsf{d}_\ast(X) := ...

**5**

votes

**1**answer

203 views

### Congruence properties of $x_1^6+x_2^6+x_3^6+x_4^6+x_5^6 = z^6$?

(This was posted previously in MSE without getting any answers.)
It is known that given primitive (co-prime) integer solutions to,
$$x_1^4+x_2^4+x_3^4+x_4^4 = z^4$$
then there is one $x_i$ such ...

**0**

votes

**2**answers

156 views

### Smallest constant so that there are at least $n/\log_2{n}$ primes between $n$ and a constant multiple of $n$

What is the smallest known $c$ so that for any $n\geq 2$ there are at least $n/\log_2{n}$ primes between $n$ and $cn$ (inclusive)?
The prime number theorem seems to give an asymptotic result so I am ...

**3**

votes

**2**answers

170 views

### Equidistribution of Hecke points and $p = (a+bi)(a-bi) = e^{i\theta}\sqrt{a^2 + b^2}$

I have seen two versions of a result called "Hecke Equidistribution" and I wanted to know if they were the same or different.
#1 Let $p = 4k+1 = (a+bi)(a-bi) = e^{i\theta}\sqrt{a^2 + b^2}$. Then ...

**1**

vote

**0**answers

69 views

### Existence of Euler product on critical line for $L(\chi,s) L(\overline{\chi},1-s)$?

Generally there is no Euler product for Dirichlet L-functions $L(\chi,s)$ in the critical strip.(cf Is the Euler product formula always divergent for 0<Re(s)<1?)
But I would like to know if ...

**1**

vote

**0**answers

46 views

### Are all complex zeros of $\Psi^{(s)}(1) \pm \Psi^{(1-s)}(1)$ on the critical line?

The balanced polygamma function $\Psi^{(s)}(x)$ for $x=1$ can be expressed as:
$$\Psi^{(s)}(1)=\dfrac{\big(\Psi(-s)+\gamma\big)\,\zeta(s+1)+\zeta'(s+1)}{\Gamma(-s)}$$
Note $\Psi(s)$ is the digamma ...

**-1**

votes

**2**answers

63 views

### Collecting sufficient conditions for Sorli's conjecture on odd perfect numbers

Sorli's conjecture predicts that, for an odd perfect number $N$ given in the Eulerian form $N = {q^k}{n^2}$ (where $q$ is prime with $\gcd(q, n) = 1$ and $q \equiv k \equiv 1 \pmod 4$), the condition ...

**9**

votes

**1**answer

395 views

### On a result attributed to W. Ljunggren and T. Nagell

I've read in a number of places that, building on previous work of T. Nagell, W. Ljunggren proved in 1 that the Diophantine equation
$$\frac{x^{n}-1}{x-1} = y^{2}$$
doesn't admit solutions in ...

**0**

votes

**0**answers

67 views

### Expliciting the distance between consecutive Goldbach numbers assuming it's finite

In this paper, the author shows unconditionally that at least one of the following statements holds:
i) the distance between two consecutive Goldbach numbers is finite, i.e. there exists an ...

**1**

vote

**1**answer

93 views

### Uniform convergence of infinite sum with Dirichlet characters

I would like to prove uniform convergence of function series like :
$$\sum\limits_{n=1}^{\infty} \chi(n) f(nx)$$ where $\chi$ is a primitive character and $f(x)$ a function decreasing to zero in ...

**3**

votes

**2**answers

195 views

### Upper bound on Betti numbers of an intersection of hypersurfaces (or quadrics)

I have a problem I have been stuck with since several weeks now, and yet I believe it should be easy to specialists.
Let $k$ be an algebraically closed field, $m$ and $n$ two integers. Let ...

**2**

votes

**0**answers

135 views

### Second differences of primes determined by increasing first differences: every positive even integer?

Suppose that $p(1)$ and $p(2)$ are primes. For $n > 2$, let $p(n)$ be the least prime $p$ such that $p-2p(n-1)+p(n-2)>0$. Does $p(n)-2p(n-1)+p(n-2)$ range through all the even positive ...