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

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8
votes
0answers
120 views

Does bounded-degree base extension yield Zariski-dense Mordell-Weil group?

If $d$ and $n$ are positive integers, does there exist a constant $B=B(d,n)$ with the following property? For any $n$-dimensional abelian variety $A$ over a degree-$d$ number field $K$, there is an ...
19
votes
0answers
281 views

Relaxed Collatz 3x+1 conjecture

The Collatz $3x+1$ conjecture claims that any positive integer can be eventually reduced to 1 by iterative application of the maps $x\mapsto 3x+1$ whenever $x$ is odd and $x\mapsto x/2$ whenever $x$ ...
2
votes
1answer
55 views

Effective Realization of GCD of middle binomials?

So, it is well-known that $$ \gcd \left(\binom{m}{k}\mid 1\leq k\lt m \right) = e^{\Lambda(m)}$$ which can incidentally be sparsified for prime $p$ $$ \gcd ...
-2
votes
0answers
67 views

Minimal number for sums and differences of primes

Let $\mathbb{N}$ denote the set of positive integers. For any set $X$, let ${\cal P}_{\text{fin}}(X)$ be the set of finite subsets of $X$, and let $\mathbb{P}$ be the set of prime numbers in ...
1
vote
2answers
89 views

Asymptotic of the sum of squared primes [on hold]

I have a rather simple question of number theory which I can't seem to be able to find a good reference for. I am not a specialist and I don't really know where to look. I would like to show that the ...
2
votes
0answers
46 views

Mestre-type algorithm for higher-genus curves?

Is there an analogous algorithm for genus $g>2$ curves that, given a complete set of invariants, outputs a curve with those invariants? (I'm interested in particular in $g=3$.) Any references ...
3
votes
0answers
57 views

Functional equation or analytic continuation of certain approximations to $\zeta^z(s)$?

Let $z$ be a complex number and $\omega(n)$ denote the number of distinct prime factors of the natural number $n$. I am considering the arithmetic functions $|\mu(n)|z^{\omega(n)}$ and their ...
4
votes
0answers
91 views

Negative coefficient in an almost cyclotomic polynomial

Let $a,b,c,d$ be four prime numbers. We set the polynomial : ...
2
votes
0answers
41 views

Questions about holomorphy and zeros of the symmetric power $L$-function

Let $f$ be a primitive form of an even weight $k$ for the full modular group and let $L(Sym^rf,s)$ be the symmetric $r$th $(r\geq 2)$ power $L$-function associated to $f.$ I have three questions ...
10
votes
2answers
689 views

Can I find Fermat's complete works anywhere?

I admire the mathematician very much and want to look at his writings. Is there anywhere in book or web form that has a collection of his writings?
3
votes
0answers
150 views

Probability that an integer contains no $1\bmod 4$ prime factor

$n$ represents integer variable. What is the probability that and integer contains at most $r(n)$ prime factors of form $1\bmod 4$ where $r(n)$ is a function of $\omega(n)$ (number of distinct prime ...
0
votes
1answer
72 views

Counting number of primorial factors

Denote $$P(n)=\prod_{p\in\mathsf{Primes}\leq n}p$$ signifying $n^{\mbox{th}}$ primorial. We know that $P(n)$ has approximately $n/\log2$ bits ...
6
votes
1answer
414 views

A road to inter-universal Teichmuller theory

What would be a study path for someone in the level of Hartshorne's Algebraic Geometry to understand and study inter-universal Teichmuller (IUT) theory? I know that it heavily relies on anabelian ...
7
votes
1answer
186 views

Sums of reciprocals involving divisor sums

This question was asked at MSE but never received an answer. Let $A\subset\mathbb{N}$ be a subset of the natural numbers, and let $\sigma(n)$ denote the sum of divisors of $n$. Recall that we have ...
3
votes
0answers
172 views

Conjectured new primality test for Mersenne numbers

How to prove that this conjecture about a new primality test for Mersenne numbers is true ? Definition: Let $M_{q}=2^{q}-1 , S_{0} = 3^{2} + 1/3^{2} , \ and: \ S_{i+1} = S_{i}^{2}-2 \pmod{M_{q}}$ ...
1
vote
1answer
97 views

Need an explanation of a deduction

When I was reading the paper of Winfried Kohnen, Yuk-Kam Lau and Igore E. Shparlinski (ON THE NUMBER OF SIGN CHANGES OF HECKE EIGENVALUES OF NEWFORMS), I found this result (which is Theorem 2 of the ...
12
votes
2answers
340 views

Are there open problems for primes which are known for probable primes?

Define "probable prime" (PP) to be natural $n>1$ satisfying $2^{n-1} \equiv 1 \pmod{n}$ or $n=2$. Probable primes are the union of the primes and base two pseudoprimes. This definition is much ...
7
votes
1answer
404 views

Sums of unique squares

Let $\mathbb{N}$ denote the positive integers and let $S = \{n^2: n\in \mathbb{N}\}$. For any positive integer $k$ we define $$\text{sq}(k) = |\{F\subseteq S: F\neq \emptyset, F\text{ is finite and } ...
2
votes
0answers
52 views

Statements generalizing representability of integers by binary quadratic forms to $n$-variable higher homogeneous forms?

Representing integers through the theory of binary quadratic forms is a well studied topic. We know that given $a,b,c\in\Bbb N$, based on discrimant $b^2-4ac$, we can study the representability of ...
14
votes
2answers
394 views

Number of distinct factors

Denote $\omega(m)$ to be number of distinct factors of $m$ as defined in http://mathworld.wolfram.com/DistinctPrimeFactors.html. At every $c>0$, given $n\in\Bbb N$ define $$S(n,c)=\big\{m\in\Bbb ...
7
votes
1answer
261 views

Irreducible cubics modulo primes

Is there a small finite (perhaps of cardinality two or three) collection of cubic polynomials $p_1, \dotsc, p_k \in \mathbb{Z}[x]$ such that for every prime $p$ at least one of these is irreducible?
2
votes
1answer
96 views

Odds of residue being small

Given $\mathsf{c\geq1}$, what is the probability that if you choose $\mathsf{A,B,\alpha\in\Bbb N}$ such that $\mathsf{A,B<\alpha<AB}$ holds we will have both ...
0
votes
1answer
80 views

Question about sign change of Hecke eigenvalues

I want to write a survey on the subject 'Sign changes for coefficients of symmetric power $L$-functions'. So, I browse the Web and I got some papers. I read it and I gave special interest to the paper ...
2
votes
1answer
85 views

A quadrant of residues

Assume that following inequality holds $$\mathsf{w,x,y,z<AB,AC,AD,BC,BD,CD<ABC,ABD,ACD,BCD<wx,wy,wz,xy,xz,yz}$$ with $$\mathsf{gcd(A,B)=gcd(A,C)=gcd(A,D)=gcd(B,C)=gcd(B,D)=gcd(C,D)=1}$$ ...
-2
votes
0answers
125 views

$\mathsf{GCD}$ in arithmetic progression

Given $\mathsf{M\in\Bbb N}$, pick $\mathsf{r,s,A,B\in\Bbb N}$ randomly with $\mathsf{0<r<s<A<B<M}$ satisfying $\mathsf{gcd(A,B)=1}$. Given $\mathsf{c\geq1}$, what is the probability ...
8
votes
1answer
809 views

Remark on Fermat's Last Theorem by Darmon, Diamond and Taylor

In their paper, Darmon, Diamond and Taylor remarked the following : (the previous paragraph of Section 2.2 (p. 55), https://www.math.wisc.edu/~boston/ddt.pdf) If $\rho : G \rightarrow ...
8
votes
4answers
380 views

Lower bounding the multiplicative order of 2 modulo p

For $p$ prime denote by $\mathsf{ord}_p(2)$ the multiplicative order of $2$ modulo $p$. Does there exist $N > 0$ such that, for ALL primes $p$, $\mathsf{ord}_p(2)$ is at least $\frac{(p-1)}{N}$? ...
2
votes
0answers
83 views

A factorial related statement

Is statement $\mathsf{S}$ below in $\mathsf{NP}$ or in $\mathsf{coNP}$? $$\mathsf{S}:\mathsf{Given}\mbox{ }n,a,s,c\in\Bbb N,\mbox{ }\mathsf{with}\mbox{ }n\mbox{ }\mathsf{a}\mbox{ }\mathsf{prime}\mbox{ ...
5
votes
1answer
314 views

A question on Ramanujan's $1/\pi$ formulas

It is known that Ramanujan discovered a number of formulas fo $1/\pi$. All of these formulas are of the form $$\frac{1}{\pi}=\sum_{n=0}^{\infty}\frac{(1/2)_n(s)_n(1-s)_n}{(1)_n^3}(a+bn)z^n,$$where ...
9
votes
2answers
406 views

Learning the exponents in a sum of two modular roots of unity

$\newcommand{\Z}{\mathbb{Z}}$ Suppose that $n$ is a large and known integer (say, with 100 digits) and that you are given access to a function $$f(x) = x^a + x^b$$ with unknown exponents $a,b \in ...
1
vote
0answers
90 views

Rank of the Jacobian of a family of hyperelliptic curves of genus 2

Assume tha $C$ be the hyperelliptic curve $y^2 = (x-a_1)\cdots (x-a_5)$ of genus $g=2$ and $a_i \in \mathbb{Z}$ and we know that the integers $a_i$ has the form $a_i= d_1^2 - d_i^2$ for some positive ...
1
vote
0answers
48 views

Avoiding the range of a bivariate integer function or Diophantine function [on hold]

I have a bivariate integer function $f(x,y)=5+23x+7y+30xy$ where $x,y \geq 0$ and are integers. The lattice points of this function, or its range contain a large number of values. I'm trying to see if ...
2
votes
2answers
268 views

Trivial zeroes of the Riemann Zeta function are simple

The trivial zeroes of the Riemann Zeta function are located on $-2\mathbb N^*$ and they are simple. It is not difficult to see that, but the proof I have in mind is using the fact that ...
-2
votes
0answers
111 views

Elementary question of Group cohomology [closed]

Let $G$ be a finite group. Assume $G$ acts on finite abelian module $M$ such that $(|G|,|M|)=1$. Question: Why $H^i(G,M) = 0$ for $i > 0$? Pierre MATSUMI
4
votes
3answers
365 views

Bateman-Horn, continued even further

As before, consider the "singular series", which shows up in the Bateman-Horn conjecture: for an irreducible polynomial $f,$ this is equal to $$ s(f) = \prod_p \frac{1-\frac{n_f(p)}p}{1-\frac1p}, $$ ...
1
vote
0answers
91 views

System of congruences

I have a system of $n$ congruences. the generic $m$_th congruence of the system ($m = 1,\dots,n$) is in the form: $(p_n\#)^2(\displaystyle\sum_{\substack{i=1\\i\neq ...
3
votes
0answers
121 views

Does the Bombieri-Lang conjecture imply severe restrictions on rational points on twists of hyperelliptic curves?

According to Silverman, the Bombieri-Lang conjecture implies that the rational points of surface on general type lie on finite set of curves, except for a finite set of points. Let $f$ be univariate ...
2
votes
0answers
61 views

Listing all Lattice Points in a Box

Let $B := [-1,1]^n$ be an $n$-dimensional box. Moreover, let $v_1,\ldots,v_n \in \mathbb{R}^n$ form a basis of $\mathbb{R}^n$, where the entries of the $v_i$ are explicitly irrational. We can assume ...
2
votes
0answers
115 views

$\frac{1}{2}<\sigma<1$, is $f(n) = \Bigl| \,1+ \frac{1}{2^{\sigma + i n}} + \cdots + \frac{1}{n^{\sigma + i n}} \Bigr|$ from $O(\log n)$?

We have $\frac{1}{2} < \sigma < 1$ and $$ f(n) = \Bigl|\, 1+ \frac{1}{2^{\sigma + i n}} + \cdots + \frac{1}{n^{\sigma + i n}} \Bigr| $$ . My goal is proving this statement that $|f(n)|$ is ...
5
votes
0answers
161 views

Congruences involving binary forms and primes of the form $x^2+y^2$

Let $a_s$ be \begin{align*} a_s=\sum_{k=0}^s{s+k\choose k}2^k, \end{align*} which is the coefficient of $x^s$ in \begin{align*} \frac{3-\sqrt{1-8x}}{2(x+1)\sqrt{1-8x}}. \end{align*} ( see ...
10
votes
3answers
424 views

Continued Fractions from Digit Streams

let $x=\sum_{i=1}^{\infty}\delta_i2^{-i},\ \delta_i\in\{0,1\}$. Is there an algorithm that converts the sequence $(\delta_0,\ \delta_1,\ ...)$ of the binary digits of $x$ to the sequence ...
0
votes
0answers
60 views

Bounds on sum of reciprocal of logarithm of primes [duplicate]

Are upper/lower bounds known for the following quantity? $$S(n,a)\stackrel{\triangle}{=}\sum_{p_k \leq n}\frac{1}{(\log p_k)^a}.$$ I am mainly interested in the case, $a=1$. I suppose with the ...
10
votes
2answers
302 views
+300

2-dimensional sublattices with all vectors having very big square (in absolute value)

QUESTION: Let $\Lambda\times\Lambda\rightarrow {\Bbb Z}$ be a lattice, that is, ${\Bbb Z}^n$ with a non-degenerate integer quadratic form, not definite, not necessarily unimodular, $n>2$. I want ...
0
votes
0answers
58 views

Smallness in modular condition

Given coprime $\mathsf{a_1,a_2\in\Bbb N}$ with $\mathsf{\mathsf{\max(a_1,a_2)\leq2\min(a_1,a_2)}}$, is there pair $\mathsf{(x_1,x_2)\in\Bbb N^2}$ such that ...
0
votes
1answer
42 views

For a given $n$, under what condition(s) there exists (at least) two different $c$ and $c′$ such that $X_n^c=X_n^{c'}$

Let $X_n^c=\{\cos\left((4k-c)\frac{\pi}{2n}+\frac{\pi}{4}\right): k=0, 1, \dots, n-1\}$ where $c\in\{0, 1, \ldots, \lfloor\frac{n}{2}\rfloor\}$ and $n$ is any positive integer greater than 3. I want ...
16
votes
2answers
757 views

Which algebraic relations are possible between algebraic conjugates?

For which non-constant rational functions $f(x)$ in $\mathbb{Q}(x)$ is there $\alpha$, algebraic over $\mathbb{Q}$, such that $\alpha$ and $f(\alpha) \neq \alpha$ are algebraic conjugates? More ...
7
votes
1answer
192 views

Numerically double-checking formula with L-values

I'm working with a special case of Ichino's triple product formula, which for classical holomorphic newforms $f$, $g$ ,$h$ of weights $k$, $m-k$, $m$ (and central characters satisfying $\chi_f \chi_g ...
2
votes
0answers
152 views

Morphism of Shimura varieties and differential equations

Is there a way of constructing a morphism between Shimura varieties using differential equations? Maybe, this looks like a completely ridiculous question, so I think that I should explain the context ...
8
votes
2answers
501 views

Does the Galois group of a Pisot polynomial contain the alternating group?

Let $n \in \mathbb{N}$, and let $p(X) \in \mathbb{Z}[X]$ be a monic polynomial of degree $n$. Suppose that exactly one complex root of $p$ is of modulus $> 1$, and that the remaining $n-1$ roots of ...
7
votes
1answer
301 views

Bounded gaps between primes in arithmetic progressions

Has Zhang's work on bounded gaps between primes been extended to the following theorem? For any arithmetic progression $an+b,\gcd(a,b)=1$, there is a constant $H$ (depending only on $a$) such that ...