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

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12
votes
4answers
3k views

Are the nontrivial zeros of the Riemann zeta simple?

A few years ago, I found on arXiv an article in which the authors (I think they were at least two to write it) claimed to have proven that the non trivial zeros of the Riemann zeta function were all ...
7
votes
1answer
320 views

Lattice points near a curve

Bombieri and Pila had a well known bound for the count of lattice points on an algebraic curve in the plane. Does it generalize to a bound for the count of lattice points near (say within a distance ...
1
vote
0answers
44 views

More generalized RSA construction

Is there a way to construct RSA type cryptosystem over general number rings? Can Number Field Sieve technique be applied here?
0
votes
1answer
300 views

Q re Kaprekar's fixed mapping points

Jens Kruse Andersen in his comment in OEIS's A099009 noticed 3 families of numbers among Kaprekar's fixed mapping points (otherwise known as kernels of the Kaprekar's routine): "Let $d(n)$ denote ...
1
vote
2answers
238 views

Is it true that $\Phi_n(2)$ has a divisor of the form $kn+1$ for all $n\neq 6$?

Let $\Phi_n(x)$ be the $n$ th cyclotomic polynomial. I've checked the values of $\Phi_n(2)$ for some small $n\geq 2$ and noticed that there is always a divisor of $\Phi_n(2)$ of the form $kn+1$ ...
2
votes
0answers
72 views

Modified Jacobi’s theta function

Be $t\in\mathbb{R}_0^+$. Jacobi’s theta function is $$\Theta(t):=\sum\limits_{k=-\infty}^{+\infty} e^{-\pi k^2 t}$$ with $$\Theta(\frac{1}{t})=\sqrt{t}\Theta(t)$$ Therefore $$\sum\limits_{k=1}^\infty ...
2
votes
1answer
256 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 $$\mathsf{gcd(A,B)=gcd(A,C)=gcd(A,D)=gcd(B,C)=gcd(B,D)=gcd(C,D)=1}$$ ...
13
votes
0answers
281 views
+50

Checking Mertens and the like in less than linear time or less than $\sqrt{x}$ space

Say you want to check that $|\sum_{n\leq x} \mu(n)|\leq \sqrt{x}$ for all $x\leq X$. (I am actually interested in checking that $\sum_{n\leq x} \mu(n)/n|\leq c/\sqrt{x}$, where $c$ is a constant, and ...
5
votes
1answer
141 views

Arbitrarily many primes in a Fibonacci-type sequence

It is conjectured that the standard Fibonacci sequence contains infinitely many primes. While this is perhaps too difficult, I am wondering about the following simpler version: Question. For any $K$, ...
2
votes
1answer
131 views

Self-containing trees

Suppose that $r^2-r-1=0$ and that $T$ is the tree with root $1$ such that the children of each node $x$ are $rx$ and $x+1$. Remove all duplicates as they occur, and let $T(r)$ denote the remaining ...
8
votes
0answers
152 views

Computing endomorphism rings of supersingular elliptic curves

I would like to know what algorithms there are to compute the linearly independent generators $(1,i,j,k)$ for quaternion algebra containing the endomorphism ring of a supersingular curve. The curve in ...
3
votes
0answers
319 views
+50

Finding the number of rational points effectively

Consider $\# P$ and $\oplus P$. There is a $\# P$-hard problem: to find number of rational solutions of a system of polynomial equations over $\mathbb{F}_2$. The corresponding $\oplus P$-hard ...
1
vote
1answer
130 views

On even almost perfect numbers other than powers of two

(Note: This question is an improved version of and has been cross-posted from this MSE post.) Let $\sigma(x)$ denote the sum of the divisors of $x$. If $\sigma(x) = 2x - 1$, then we call $x$ an ...
6
votes
2answers
527 views

Generalizing Ramanujan's “1729 story”

Whenever I read the anecdote about Hardy, Ramanujan and the taxi number 1729 I'm amazed that it could have occurred to anyone just off the top of their head that 1729 can be written as the sum of two ...
0
votes
0answers
183 views

On the coherence of formal power series ring

Let $A = {\Bbb F}_p[[X_1,X_2,...]]$ be the ring of formal power series with infinitely many variables over the finite field ${\Bbb F}_p.$ $A$ consists of such formal sum elements as $\sum ...
11
votes
2answers
751 views

Tate uniformization of nonsplit semistable elliptic curves

Let $E/\mathbf{Q}_p$ be an elliptic curve having split multiplicative reduction. Then Tate uniformization gives a surjective homomorphism of $p$-adic analytic groups $G_m \to E$, with infinite cyclic ...
1
vote
0answers
54 views

For which primes $p$ is the field $\mathbb{Q}(\Gamma(1/p^{j}))$ a strict subfield of $\mathbb{Q}(\Gamma(1/p^{i}))$ whenever $0<i<j$?

I already asked this question on a French math forum but eventually came to think that as silly it may turn out to be, perhaps something interesting could finally emerge from it, so I decided to take ...
4
votes
1answer
229 views

Does anyone recognize this exponential sum?

For $a$, $b$ two integers, let $(a,b)$ denotes their gcd. We define the following exponential sum : $$G_q(n):=\sum_{d|q,~(d,q/d)=1}{e^{2i\pi n\frac{dd'}{q}}}$$ for $n$ a non-negative integer and $q$ ...
0
votes
2answers
131 views

Function that gives 1 only when an integer is divisible by another integer [on hold]

I need a function that takes two inputs, a and b, and returns 1 only when a is divisible by b and 0 otherwise. Can this be written in a nice mathematical way (other than using indicator functions)?
3
votes
1answer
400 views

origin of analogy “primes as the atoms of number theory/ arithmetic”

a math student recently challenged me on the old comparison/ analogy of prime numbers to "the atoms of number theory or arithmetic" and then was wondering the origin of the phrase. where does this ...
0
votes
0answers
110 views

A number theory question [duplicate]

Let $m,n$ be integers and $P$ be a prime and $\lambda$ be an integer such that $0 < \lambda <\sqrt p$, Can the following equation have any answer for any $m,n,p,\lambda$ except $\lambda=+-1$? ...
4
votes
1answer
103 views

Nearly Be Bruijn sequences constructed from De Bruijn sequences

Let $w$ be a De Bruijn $01$-sequence of the type $B(2,n)$; i.e., a cyclic $01$-sequence that contains every $n$-digit $01$-sequence exactly once. Let $x$ be a $01$-sequence of length $n$. When and ...
0
votes
0answers
87 views

A number theory question related to algebraic graph theory? [on hold]

Let $m,n$ be integers and $P$ be a prime and $\lambda$ be such that $0 < \lambda <\sqrt p$, Can the following equation have any answer for any $m,n,p,\lambda$? ...
2
votes
1answer
64 views

On cluster points of a particular sequence

This is the sequel of a previous question. Let us consider the sequence $$ \xi_n = 2n \{n\xi\}-n, $$ where $\xi>0$ is a given real irrational number and $\{\cdot\}$ is the fractional part. Do ...
-2
votes
2answers
159 views

Precise asymptotic of diophantine approximation

I know that every irrational number $\xi$ can be approximated by rational numbers in such a way that $$ \left| \xi - \frac{p}{q} \right| \leq \frac{c}{q^2} $$ for infinitely many choices of $p$ and ...
5
votes
2answers
510 views

Is the Crandall, Dilcher and Pomerance heuristic concerning Wall-Sun-Sun primes still state of the art?

This is a question about the open problem Fibonacci divisibility from the Open Problem Garden. The problem, originally stated in 1960 by D.D. Wall, has several equivalent formulations one of which ...
2
votes
1answer
173 views

Number of rational points in a non-smooth variety

Let $X$ be an algebraic variety over $\mathbb{F}_q$ with dimensional $n$. We know that if $X$ is smooth than $X$ has about $q^{nk}$ rational points over $\mathbb{F}_{q^k}$ (Weil hypothesis). Is there ...
0
votes
0answers
68 views

How to show this bound? [closed]

Let $f$ be a primitive of an even weight $k\geq 2$ for the full modular group and denote $\lambda_f(n)$ its $n$-th normalized Fourier coefficient. Can someone provide me with explicit constants for ...
20
votes
1answer
1k 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 $2^n \pm 1$? We recall that radical of an integer $rad(k)$ is a product of primes which divide $k$. As an example, if ...
4
votes
1answer
339 views

Index of the Hecke algebra with operators omitted

This is a spin-off to the question Omitting primes from a Hecke algebra by David Loeffler. Let $N$ be a positive integer. For a finite set of primes $\Sigma$, let $\mathbb T^{\Sigma}$ be the $\mathbb ...
8
votes
3answers
1k views

Is there a schemetical construction for modular curves over the rationals?

One can get modular curves by the following procedures: first take the uper half plane and the rationan numbers on the x-axis, then we consider the quotient by a congruence subgroup. Now we get a ...
3
votes
1answer
121 views

Generalized Equal Distribution Kolakoski Sequence Conjecture

If we let $\operatorname{Kol}(a_1,\dots,a_n)$ be the run sequence determined by the rules of Kolakoski Frequencies, we ask is there a sequence of $\operatorname{Kol}$ that DOES NOT obey the $1/n$ ...
1
vote
1answer
107 views

is there any more odd near-perfect number?

we know the first odd near-perfect number is $3^4*7^2*11^2*19^2$(of course the number must be square).and from one paper ,which is studying the odd perfect number and giving some estimate on it, we ...
15
votes
2answers
627 views

A set of integers whose factorial can be written as a product of two factorials

I am trying to collect informations concerning the set $$\mathcal{A}=\left\{n\in\mathbb{N} \mid (\exists k,l\in\{2,3,\dots,n-2\})(n!=k!l!)\right\}.$$ It seems not much is known about the set ...
0
votes
1answer
92 views

Algorithm to check if a number is the sum of another number and its reverse [closed]

So im looking for an algorithm that checks (in about 10 sec) if a natural number M (1≤M≤10^100000 -yes, the range is that big) can occur by the sum of another natural number N and its reverse Nr. For ...
14
votes
4answers
758 views

Rational solutions of the Fermat equation $X^n+Y^n+Z^n=1$

As a generalisation to the equation of Fermat, one can ask for rational solutions of $X^n+Y^n+Z^n=1$ (or almost equivalently integer solutions of $X^n+Y^n+Z^n=T^n$). Contrary to the case of Fermat, ...
0
votes
0answers
108 views

A reflection on the Fermat equation

$xy(x+y)^n=1$ with $n\in\mathbb{N}$ without any solution for $x,y\in\mathbb{Q}^+$ is equivalent to (and therefore a trivial generalization of) the Fermat Theorem $a^{n+2}+b^{n+2}=c^{n+2}$ without any ...
5
votes
2answers
155 views

Mahler measure of a totally positive, expanding algebraic integer

Consider a degree-$d$ algebraic integer $\alpha$ all of whose conjugates (including itself) are real numbers greater than 1. Its Mahler measure $M(\alpha)$ is simply equal to the norm $N(\alpha)$. ...
-1
votes
1answer
167 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 ...
15
votes
1answer
360 views

What do Hecke eigensheaves actually look like?

Let $\mathbb F_q$ be a finite field, $C$ a curve over $\mathbb F_q$ of genus $g\geq 2$, $\rho: \pi_1(C) \to GL_2(\overline{\mathbb Q}_\ell)$ an irreducible local system. The geometric Langlands ...
1
vote
1answer
168 views

Finding cyclic subgroups of points on elliptic curves for isogeny based cryptography

Isogeny based cryptography is one of the newest post-quantum cryptography. Hardness of this system is based on finding isogeny between two elliptic curves. Also this is a theorem: Elliptic curves ...
4
votes
2answers
255 views

The best possible density in Hilbert's Irreducibility Theorem

Let $f(X,t_1,\dots,t_s)$ be an irreducible polynomial with coefficients in $\mathcal{O}_K$, the ring of integers of a number field $K$. By work of S. D. Cohen ...
3
votes
0answers
175 views

An explicit formula for $\zeta(2m+1)$ with good convergence

The question: Is the following formula known? $$\zeta(2m+1)=\frac{(-1)^m 2^{4m+2}\pi^{2m}}{2^{2m}-1} \sum\limits_{k=1}^m \frac{(2^{2k}-1)b_{2k}}{2^{2k}(2k)!} \sum\limits_{v=k}^m ...
1
vote
1answer
137 views

Powers of two with coefficients {1,−1}

Given a vector $(n_0, n_1, \dots, n_l)$ where $n_i \in \{-1, 1\}$, $i = \overline{0, l-1}, n_l = 1$ and $l \in \mathbb{N}$. Prove that for all $a$ such that $$0 < a \leq 2^0\cdot n_0 + 2^1 \cdot ...
25
votes
3answers
927 views

$\zeta(n)$ as a mixed Tate motive

I am trying to understand why there exists, for each $n \geq 2$, a mixed Tate motive $M$ over $\mathbb{Q}$ such that $M \in Ext^1_{MT(\mathbb{Q})}(\mathbb{Q}(0), \mathbb{Q}(n))$ and $\zeta(n)$, ...
7
votes
1answer
190 views

If two Hecke characters cut out the same field, are they Galois conjugates?

First question on MathOverflow, I hope it is appropriate for this site. There are two related questions. Let $K$ be a number field, $G_K = Gal(\overline{K}/K)$, $p$ a prime, and ...
14
votes
3answers
2k views

Are there pairs of consecutive integers with the same sum of factors?

Background/Motivation I was planning to explain Ruth-Aaron pairs to my son, but it took me a few moments to remember the definition. Along the way, I thought of the mis-definition, a pair of ...
5
votes
1answer
304 views

Unique quadratic subextension of a ray class field

Let $K_q$ denote the unique quadratic subextension of the ray class field over $\mathbb{Q}$ of conductor $q\times\infty$. Then $K_q$ should be $\mathbb{Q}(\sqrt{q})$ if $q$ if 1 mod 4 and ...
2
votes
0answers
72 views

How to estimate $\prod_{t=1}^{N}\frac{1}{2-z^t}$ for large $N$?

Based on the top answer to How to estimate of $\prod_{k=a}^N \frac{1}{e^{k\kappa}-1}$ for large $N$? Can anyone find an approximate closed form for $$ ...
1
vote
0answers
76 views

Are there only finitely many fixed degree nontrivial polynomial parametrizations of the surface $x^4+y^4=z^4+t^4$?

Consider the surface over the rationals $$ x^4+y^4=z^4+t^4 \qquad (1)$$ Consider parametrizations of form: $$ f_1(u)^4+f_2(u)^4=f_3(u)^4+f_4(u)^4 \qquad(2) $$ where $f_i$ are polynomials with integer ...