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

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11
votes
0answers
127 views

What can topological modular forms do for number theory?

Topological modular forms ($TMF$) have in the recent years made an impact in algebraic topology. Roughly, the spectrum $tmf$ is the (derived) global sections of the sheaf of $E_\infty$ ring spectra ...
1
vote
1answer
143 views

How does the solenoid structure of $\mathbb{A}/\mathbb{Q}$ lift to $PGL(2, \mathbb{A})/ PGL(2, \mathbb{Q})$?

Some papers I am reading talk about an "adelic" object $PGL(2, \mathbb{Q}) \backslash PGL(2, \mathbb{A})$ . This has sparked a lot of confusion since I don't know what such a quotient could mean. A ...
5
votes
2answers
154 views

Sumsets and dilates: does $|A+\lambda A|<|A+A|$ ever hold?

The following problem is somehow hidden in this recently asked question, but I believe that it deserves to be asked explicitly. Is it true that for any finite set $A$ of real numbers, and any real ...
5
votes
0answers
101 views

Rankin-Selberg for Maass form GL(3)xGL(2)

Let $F$ be a Maass cusp form for $\mathrm{SL}(3,\mathbb{Z})$ (level 1 trivial character). Let $g$ be a Maass cusp form for $\Gamma_0(N)$ with character $\chi$ mod $N$. For convenience, you may assume ...
0
votes
0answers
77 views

Four kinds of generalized hypergeometric formulas for $\pi$

Given, $$\begin{array}{|c|c|c|c|} \hline n&a_n&b_n&c_n\\ \hline 1 & 6541681608 & 163096908 & -640320^3\\ \hline 2 & 85840 & 4492 & -14112^2\\ \hline 3 & 28302 ...
0
votes
0answers
93 views

Does the equality of product of integers modulo prime p holds in a given interval?

For any given prime $p$, does there exist $a_1,a_2,\dots,a_k,$ (not necessarily distinct) $b_1,b_2,\dots,b_m$ (not necessarily distinct) and $y_1$, $y_2$ such that ...
1
vote
0answers
43 views

What is the complexity of finding a generator for the cyclic elliptic curves?

Let $E$ be an elliptic curve which is defined over a finite field $\mathbb{F}_p$, where $p$ is a prime number. If we know that $E(\mathbb{F}_p)$ is cycyclic, is there an algorithm to find its ...
0
votes
0answers
37 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 ...
3
votes
0answers
108 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
vote
1answer
90 views

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

(Note: This has been cross-posted to MSE. However, I feel that it is more likely to receive a good answer here, because I believe that it is a research-level question. For the mathematicians who ...
0
votes
0answers
105 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
0answers
59 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 ...
10
votes
3answers
373 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 ...
13
votes
1answer
497 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
1answer
117 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
173 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
0answers
66 views
-1
votes
0answers
35 views

Implementation of almost integer to cryptography [closed]

Can there be any implementation of almost integers to create intractable problems relevant to public key cryptography?
5
votes
0answers
62 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
0answers
171 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
0answers
43 views

Periodicity of any fermat number modulo a prime [closed]

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
0answers
143 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 ...
40
votes
4answers
1k 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
3answers
306 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
2answers
222 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
1answer
165 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
1answer
102 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
0answers
62 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
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 $L=Q$? ...
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 ...
4
votes
3answers
184 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
162 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
0answers
78 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
0answers
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
1answer
96 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
117 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, ...
4
votes
1answer
157 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
3answers
557 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
311 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
2answers
508 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
2answers
932 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
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 ...
3
votes
0answers
154 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
2answers
345 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
1answer
133 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
1answer
204 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
2answers
157 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
2answers
174 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
0answers
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
0answers
48 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 ...