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

learn more… | top users | synonyms (1)

2
votes
0answers
63 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 ...
2
votes
0answers
107 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 $M_{q}=2^{q}-1 , S_{0} = 3^{2} + 1/3^{2} , \ and: \ S_{i+1} = S_{i}^{2}-2 \pmod{M_{q}}$ ...
12
votes
5answers
12k views

Enumerating ways to decompose an integer into the sum of two squares

The well known "Sum of Squares Function" tells you the number of ways you can represent an integer as the sum of two squares. See the link for details, but it is based on counting the factors of the ...
10
votes
2answers
686 views

Most dense subset of numbers that avoids arbitrarily long arithmetic progressions

The famous Green-Tao theorem says that there exist arbitrarily long sequences of primes in arithmetic progression. I am wondering: How dense can a subset $S \subset \mathbb{N}$ be and still avoid ...
12
votes
2answers
281 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 ...
15
votes
1answer
776 views

Higher level analogs of Nicolas-Serre theory

NICOLAS-SERRE THEORY Let $F \in Z/2[[x]]$ be $x+x^9+x^{25}+...$, the exponents being the odd squares, and $V$ be the space spanned by the $F^k$ with $k$ odd. Nicolas and Serre define formal Hecke ...
0
votes
0answers
151 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 ...
6
votes
1answer
140 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 ...
10
votes
2answers
267 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 ...
2
votes
2answers
270 views

Primality criteria for specific class of Wagstaff numbers ?

I asked this question on mathstackexchange but didn't get any answer . Definition : Let $W_p$ be a Wagstaff number of the form : $W_p=\frac{2^p+1}{3}$ , with $p\equiv 1 \pmod 4$ Next , ...
7
votes
1answer
345 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 } ...
1
vote
1answer
80 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 ...
8
votes
1answer
797 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 ...
7
votes
4answers
363 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
1answer
233 views

Two questions about Whittaker functions

I am watching the video: Modeling p-adic Whittaker functions, Part I. I have two questions about Whittaker functions in the video. From 33:00 to 37:00, it is said that after changing of variables, ...
14
votes
2answers
364 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 ...
2
votes
1answer
90 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 ...
19
votes
4answers
2k views

Covering $\mathbb{N}$ with prime arithmetic progressions

For every prime $p_i>2$ choose a $k_i\ge p_i$ , $k_i \in \mathbb{N}$ and take the arithmetic progression $A_i=k_i+np_i$ $n \ge 0$ . Is there any choice of the $k_i's$ such that $|\mathbb{N} ...
2
votes
1answer
83 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 $$\mathsf{gcd(A,B)=gcd(A,C)=gcd(A,D)=gcd(B,C)=gcd(B,D)=gcd(C,D)=1}$$ ...
4
votes
3answers
360 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 $$ s(f) = \prod_p \frac{1-\frac{n_f(p)}p}{1-\frac1p}, $$ ...
2
votes
0answers
48 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 ...
42
votes
4answers
3k views

If the Riemann Hypothesis fails, must it fail infinitely often?

That is must there either be no non-trivial zeros off the critical line or infinitely many? I'm sure that no one believes otherwise, but I've never seen a theorem in the literature addressing this. ...
7
votes
1answer
254 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?
0
votes
1answer
76 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 ...
3
votes
3answers
1k views

Tate module of CM elliptic curves

This is an exercise in Silverman's book "the arithmetic of elliptic curves". Ex 3.24, page 109. E/K CM elliptic curve, Prove for $\ell \neq char(K)$, the action of $Gal(\bar{K}/K)$ on $T_{\ell}(E)$ ...
-2
votes
0answers
123 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 ...
10
votes
3answers
422 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 ...
5
votes
0answers
263 views

Questions (related to deformation theory?) about modular ideals in mod ell Hecke algebras

I suspect that I'm asking (familiar?) questions from deformation theory in a different language. But I'm an illiterate in deformation theory language; if my suspicion is correct I'd be grateful for an ...
34
votes
1answer
2k views

Optimization problem arising from the study of zeta zeros

Motivation: The following problem arose in [1] while studying the vertical distribution of the zeros of the Riemann zeta-function. At the time, my collaborators and I were unable to solve it and I ...
2
votes
3answers
677 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.
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
309 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 ...
0
votes
1answer
168 views

Chinese remainder theorem for cyclic subfactor planar algebras

This post was inspired by an exchange with the indian woman mathematician Ajit Iqbal Singh. The chinese remainder theorem can be stated as follows: Let $n_1, \dots, n_r \ge 2$ be positive integers ...
2
votes
1answer
293 views

Quadratic Diophantine equation in $\mathbb Z[T]$

I am trying to solve the following quadratic diophantine equation in $\mathbb Z[T]$: $$((T+1)X+TY-1-Z)((T+1)X+TY-1+Z)=24XY$$ One has the following trivial solutions: $(X,Y,Z)=(0,Y,\pm(1-TY))$, ...
9
votes
2answers
401 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 ...
2
votes
2answers
261 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 ...
1
vote
0answers
88 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
46 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 ...
4
votes
2answers
200 views

degeneration of reductive group

If $A$ is a mixed characteristic complete DVR (I'm only actually interested in $\mathbf{Z}_p$) and $G/A$ is a closed subgroup scheme of $GL(n)$ whose generic fibre is connected reductive and split, is ...
-2
votes
0answers
107 views

Elementary question of Group cohomology [on hold]

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
16
votes
2answers
747 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 ...
9
votes
1answer
422 views

Uniformly small sums of roots of unity

I have considerable numerical evidence that for all $0\leq k\leq{{n-1}\over 2}$ ($n$ odd) there exists a subset $ S_k$ of {1,2,...,n} of cardinality $k$ such that the modulus square of ...
1
vote
0answers
88 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
117 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 ...
8
votes
1answer
490 views

Estimate on radical of $2^n \pm 1$

Not sure if this belongs to MO or not. Are there any lower bound on radical of $2^n \pm 1$ (Recall that radical of integer $rad(k)$ is a product of primes which divide integer $k$)? As an example If ...
2
votes
0answers
60 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 ...
4
votes
0answers
796 views

integer solutions of $ (n!+1)=m^2$

Consider $4!=24$, if you add one you get $25=5^2$. The same occurs with $5! = 120 = 11^2 - 1$, and $7! = 5040 = 71^2 - 1$. Are there other solutions of the equation $n!+1 = m^2$? I verified that no ...
2
votes
0answers
110 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
157 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 ...
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 ...