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

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6
votes
1answer
133 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 ...
4
votes
0answers
440 views

An Expression for $\log\zeta(ns)$ derived from the Limit of the truncated Prime $\zeta$ Function

I think, here, I found $$ P_x(s)=\sum_{p < x} \frac{1}{p^s} =\sum_{n=1}^{\infty}\frac{ \mu (n)}{n} \sum_{z\in\{1,\rho\}}(-1)^{1-\delta_{1z}} \left[ {\rm li}(t^{\frac zn-s}) \right]^{x}_2 \tag{7} ...
8
votes
0answers
248 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 $$ P(x_1, \dots, x_k) = 0, \ \ \ P \in \mathbb{Z}[x_1, \dots, x_k] $$ ...
4
votes
1answer
228 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 $a_i = \lfloor \alpha_i \rfloor$, ...
5
votes
1answer
156 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 ...
57
votes
5answers
22k views

Consequences of the Riemann hypothesis

I assume a number of results have been proven conditionally on the Riemann hypothesis, of course in number theory and maybe in other fields. What are the most relevant you know? It would also be nice ...
1
vote
0answers
78 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
0answers
155 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 ...
38
votes
4answers
945 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 ...
3
votes
2answers
218 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 ...
-1
votes
0answers
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
0answers
52 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 ...
-4
votes
0answers
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 ...
-5
votes
0answers
158 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 ...
10
votes
2answers
502 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 ...
4
votes
3answers
300 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 ...
-1
votes
0answers
137 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 ...
2
votes
1answer
98 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) ...
4
votes
0answers
212 views

Analog of Euler's factoring technique

Is there an analog of Euler's Two Squares factoring theorem over polynomial rings $\Bbb Z[x]$ by considering a version for non-negative polynomials? Euler's two squares factoring states that numbers ...
6
votes
1answer
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 $L=Q$? ...
6
votes
0answers
58 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 ...
2
votes
1answer
178 views

Siegel-Walfisz for the Möbius function

I am working through the proof of the Bombieri-Vinogradov theorem in Analytic Number Theory (Iwaniec, Kowalski). My problem is that on page 424, it is said that $\mu(m)$ satisfies $D_f(x;q,a)\ll ...
1
vote
0answers
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
3answers
181 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
0answers
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 ...
12
votes
6answers
2k views

On Polynomials dividing Exponentials

EDIT, May 2015: in the second edition of the relevant book, the question was corrected, a single number had been mis-typed. The corrected question (thanks to Max) is to find all positive integer pairs ...
6
votes
1answer
174 views

Prescribed values for the uniform density

Strauch & Tóth [1] Georges Grekos [3][4] showed that for any choice of upper and lower density, there is some subset of $\mathbb{N}$ with the chosen densities, provided the lower is no more than ...
3
votes
0answers
71 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, ...
0
votes
0answers
295 views

Looking for product of symmetric polynomials evaluated at roots of unity

Consider $a_{1} = \alpha^{N-n}, a_{2} = \alpha^{N-n+1}, a_{3} = \alpha^{N-n+2}, a_{4} = \alpha^{N-n+3}, \cdots, a_{n} = \alpha^{N-1}$ where $\alpha$ is a complex $N$th root of unity where $N = 2 + ...
6
votes
2answers
283 views

Minimum number of unit fractions to sum up a given positive rational

For any positive $p,q\in\mathbb{N}$ there is a finite subset $S$ of $\{\frac{1}{n}:n\in\mathbb{N}, n\geq 1\}$ such that $\sum_{s\in S} s=\frac{p}{q}$, see this article by Paul Erdös and Sherman Stein ...
-1
votes
0answers
63 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
1answer
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
0answers
113 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, ...
9
votes
1answer
648 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 ...
2
votes
3answers
543 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
1answer
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 ...
3
votes
1answer
149 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 ...
11
votes
1answer
375 views

Transcendence of products of certain real algebraic numbers

Let \begin{equation} z := \prod_p p^{1/p^2}, \end{equation} where the product is over all prime numbers $p$, and we always take the positive real root. Is $z$ transcendental or algebraic, or (as I ...
11
votes
2answers
910 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?
3
votes
2answers
340 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 $\left(\frac{k+1}{p}\right)$ ...
13
votes
3answers
1k views

Diophantine equation: Egyptian fraction representations of 1

According to the OEIS (A002966) there are 294314 solutions in positive integers to the equation $$\sum_{i=1}^7\frac{1}{x_i}=1$$ assuming $x_1\leq x_2\leq\cdots\leq x_7$. Similarly for 8 summands there ...
5
votes
1answer
144 views

A generalization of Erdős–Newman–Mirsky?

Given a sequence $S$ of natural numbers, write ${\bf Gap}(S)$ for the set of differences between consecutive terms. (So $|{\bf Gap}(S)|=1$ precisely for arithmetic progressions, hence the connection ...
5
votes
1answer
140 views

On the number of 3-Selmer elements of rational elliptic curves

I am trying to understand a step in the proof of Theorem 39 in the recent work of Bhargava and Shankar, "Ternary Cubic Forms having bounded invariants, and the existence of a positive proportion of ...
5
votes
1answer
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 ...
-1
votes
0answers
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 ...
11
votes
4answers
4k views

Fast computation of multiplicative inverse modulo q

Given a large number $q$ (say, a prime) and a number $a$ between 2 and $q-1$ what is the fastest algorithm known for computing the inverse of $a$ in the group of residue classes modulo $q$?
6
votes
3answers
408 views

Conditions for an analogue of Cauchy-Davenport for simple groups

What conditions would be sufficient for a generalization of Cauchy-Davenport for simple groups? I can see two possible difficulties with a generalization for general groups: The sets could both be ...
4
votes
1answer
457 views

Number of solutions in a sum of squares Diophantine equation

Let $n$ be an integer. I'm interested in upper bounding the number of integer solutions of \begin{equation*} x_1^2+x_2^2+\ldots+x_p^2=y_1^2+y_2^2+\ldots+y_p^2, \end{equation*} where for all ...
9
votes
1answer
394 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 ...