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

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2
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52 views

A Collatz-like function that bifurcates on primes

This is likely piling one mystery on another, but ... I was exploring a function $f(n): \mathbb{N} \mapsto \mathbb{N}$ defined as follows: $$ f(n) = \begin{cases} n^2 & \text{if} \;n \;\text{is ...
0
votes
0answers
34 views

Twisted Padé approximants

Let $f$ be a continuous function defined on $\mathbb Z_p$. By Mahler theorem, there exists a sequence $(\gamma_k)_{k\in\mathbb N}$ of $\mathbb Z_p$ such that for every $z\in\mathbb Z_p$ ...
1
vote
0answers
108 views

A Question about Palindromic Numbers and System of Arithmetic Progression

Based from Harminc and Sotak's result, www.fq.math.ca/Scanned/36-3/harminc.pdf we know that under certain condition, an arithmetic progression can contain an infinitely many palindromes. My question ...
0
votes
1answer
106 views

Sums of two squares: positive lower density? [duplicate]

This question was (indirectly) raised in this post. A set $A\subseteq \mathbb{N}$ has positive lower density if $$\text{lim inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} > 0.$$ Does the set ...
4
votes
0answers
59 views

genus 2 Seigel theta series of 3-dimensional lattices

Let $(V,f)$ be a $3$-dimensional positive definite quadratic space over $\mathbf Q$. Let $G(V)$ be a set of representatives of the isometry classes of maximal integral lattices on $V$. To an ...
0
votes
0answers
42 views

Representations of Hamilton's real/complex quaternions algebra

A lot of works and questions deal with classifying representations of a simple central algebra of given dimension over a non-archimedean field, for instance here. But do we know precisely such a ...
2
votes
2answers
250 views

Sums of sets of lower density 0

We say that a set $A\subseteq \mathbb{N}$ has lower density 0 if $$\text{lim inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} = 0.$$ Given $A,B\subseteq \mathbb{N}$ we set $A+B = \{a+b: a\in A, b\in ...
3
votes
1answer
106 views

Simultaneous approximation by rationals with relatively prime numerators

The following seems hard to me (or perhaps just not true), but perhaps I am mistaken. It is known that given irrational numbers $x_1$ and $x_2$, there are infinitely many simultaneous rational ...
3
votes
2answers
211 views

n torsion groups of quadratic twists of elliptic curves

If $E$ is an elliptic curve over a number field $K$ and $E^{F}$ is a quadratic twist of $E$. Then it is stated in ``Ranks of twists of elliptic curves and Hilbert’s tenth problem" due to Mazur and ...
6
votes
0answers
150 views

Which irrationals yield bounded sets of iterates?

For $r > 0$, define $f(n) = \lfloor {nr}\rfloor$ if $n$ is odd and $f(n) = \lfloor {n/r}\rfloor$ if $n$ is even. For which irrationals $r$ is the set $\{n,f(n), f(f(n)),\dots\}$ bounded for every ...
0
votes
2answers
245 views

Distribution of composite numbers

Motivation: I want to give an abstract formulation of Eratosthenes's Sieve and try to conject a property of Eratosthenes's Sieve. If this property are right, then I can use it to find a low bound of ...
0
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0answers
79 views

Mod 2 modular forms in levels 5 and 25--how to account for this Hecke isomorphism?

The space $P1$ of my earlier question 203755 "Two spaces attached to mod 2 level 9 modular forms...", is essentially the space of mod 2 level 3 modular forms. That such a space should appear inside ...
2
votes
0answers
229 views

How is $ \sum_{x \in X(\mathbb{F}_q)} \dots $ a generalization of cardinality?

This quarter Maxim Kontsevich is offering a course on exponential integral. There is not much in the way of notes, but is one page with mysterious comments. Let $X$ be an algebraic variety over ...
0
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0answers
107 views

The following is a necessary condition for a number to be prime, from its digit expansion. Is it already known? [on hold]

Concerning a numbers’ digits we know some neccessary conditions on them for the number to be prime, besides the last digit having to be odd (except for prime 2). For example in decimal representation ...
3
votes
0answers
96 views

Farey Fractions Estimate Equivalent to the Prime Number Theorem?

Wikipedia's article on Farey Fractions points to an article of Jerome Franel that some averages are equivalent to the Riemann hypothesis. Let $F_n$ be the $n$-th Farey sequence, then the number of ...
2
votes
1answer
233 views

Powers modulo a fixed integer

We say that a set $A\subseteq \mathbb{N}$ has positive measure if $$\text{lim inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} > 0.$$ For $b\in\mathbb{N}$ with $b>1$ we consider the sets $$S_b ...
0
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0answers
80 views

Concentration of large prime factors of polynomials

For each positive integer $n$, let $P(n)$ denote the largest prime factor of $n$ (and for completeness, define $P(1) = 1$, say). Let $f(x) \in \mathbb{Z}[x]$ be an irreducible polynomial of degree at ...
-4
votes
0answers
63 views

Solution of the equation $M^{2n}+DM^nK^n+K^{2n}=Z^2$ [on hold]

One way to obtain solutions of this equation when $M$, $N$, $n$ are natural numbers and $n≥2$ is as follows. We set $b^n=K^{2n}*(M^n+2)+2K^n*(M^n+1)+M^n$ and $D=(b^n-M^n)/K^n$ and $Z^2=(K^{2n} ...
12
votes
0answers
506 views

Meaningful review of Moriwaki's “Arakelov Geometry”

I have been asked to write a mathscinet review for Atsushi Moriwaki's Arakelov Geometry book: http://www.ams.org/bookstore-getitem/item=mmono-244 I could do the review the standard way in a day or ...
-4
votes
1answer
153 views

Remainder is always a multiple of 9 [on hold]

I am not a mathematician and certainly number theory is not my forte, but I have found the following pattern inspired on Kaprekar's routine (I am pretty sure this is already well-known although I have ...
0
votes
1answer
150 views

Is the following conjecture equivalent to the Second Hardy-Littlewood Conjecture? [on hold]

Let $y$ be an arbitrary positive real number such that $y\ge 2$. Then if we can prove that, $$\lim_{x\to\infty}\dfrac{\pi(x)+\pi(y)}{\pi(x+y)}=1$$will it imply that for all sufficiently large $x$ ...
0
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1answer
142 views

Redundancy of the Cantor Enumeration of the Rationals

What is the cardinality of the set of values corresponding to the first $n$ rationals generated in Cantor's enumeration scheme for proofing of their countability? Edit: following the suggestion of ...
6
votes
0answers
85 views

primality and square freeness of the partition function

Divisibility properties of the partition function $p(n)$ seem to have been studied for the last three hundred years (most recently, Ken Ono has been quite active). However, I assume it is open that ...
1
vote
1answer
222 views

When can we write fundamental units explicitly

Given a number field $K$, the Dirichlet Unit Theorem tells us about the structure of the unit group $O_K^\times$. However, the proofs do not seems to give any way to explicitly write out a set of ...
9
votes
1answer
240 views

Higher Fano varieties and Tsen's theorem

The rational connectivity of (complex) Fano manifolds ($c_1(T_X) > 0$) is one of the major, and surely most memorable achievements of Mori's bend-and-break method. To this day, despite intensive ...
8
votes
1answer
289 views

Orthonormal bases of R^3 with components lying in the golden field

Greg Egan proved an interesting theorem about unit vectors in $\mathbb{R}^3$ whose components actually lie in the 'golden field' $\mathbb{Q}[\sqrt{5}]$. He found it in our studies of twin ...
1
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0answers
84 views

Divisibility in $F3[t]$

Let $(P_n)_n$ be the sequence of polynomials on $\mathbb F_3$ defined by $$P_n=\prod_{\substack{h\in\mathbb F_3[t]\\\deg\,h=n\\h\text{ monic}}}h\qquad (P_0=1).$$ For every $r\in\mathbb N^*$, ...
5
votes
1answer
135 views

Computing minimal polynomials using LLL

I would like to use the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (LLL algorithm) to compute the minimal polynomial of a (real) algebraic number $\alpha$ from a decimal approximation ...
9
votes
2answers
537 views

What are the current trends in class field theory?

Being far from an expert in the subject I was wondering if people can hint towards a modern exposition of the developments in the last 10 years ? Or if not then suggest some sub-subjects in CFT that ...
1
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0answers
39 views

Big Omega result about number of totally positive integers with fixed trace

There is much literature on the study of $N_a$, the number of totally positive integers with fixed trace $a$ in a totally real field. That number has a natural geometric approximation $G_a$, and we ...
0
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0answers
80 views

Generalized arithmetic progressions contained in Bohr sets

Recall that generalized arithmetic progression of dimension $d$ is by definition a set of the form $P = P_1+\dots+P_d$, where $P_j = \{lp_j\ \mid \ |l|\leq l_j\}\subset \mathbb Z$ is an ordinary ...
6
votes
1answer
299 views

Distribution of the sequence $\bigl (\frac{\phi(n)}{n}\bigr )_{n=1}^{\infty}$ in $[0,1]$

Maybe this is a well-know problem. What do we know about distribution of the sequence $\bigl (\frac{\phi(n)}{n}\bigr )_{n=1}^{\infty}$ in $[0,1]$? (Where $\phi$ is the Euler's totient function). In ...
1
vote
0answers
41 views

Question about existence of forms with small $h$-invariant satisfying certain property

Given a form $f \in \mathbb{Q}[x_1, ..., x_n]$ of degree $d>2$, we define $h(f)$ to be the smallest positive number $h$ such that we can write $$ f = u_1v_1 + ... +u_h v_h, $$ where each $u_i$, ...
4
votes
0answers
84 views

Integer decomposition of dilated integral polytopes

For $n > 0$, let $P$ be an integral polytope, that is, the convex hull in $\mathbb{R}^n$ of points in $\mathbb{Z}^n$. Suppose that $\dim(P) = n$. Question: Given $d > n + 2$ is it true that $$ ...
5
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0answers
92 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 ...
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votes
0answers
239 views

When do boring objects exist? [closed]

Let's provisionally call an integer boring if it is not the root of a polynomial over $\mathbb{Z}$ with a small number of variables and with small coefficients and arguments (note that this requires ...
1
vote
3answers
266 views

Has this formula for $G_{k}:=\lim\inf_{n\to\infty}p_{n+k}-p_{n}$ been conjectured?

I give here a heuristics that suggests that the quantity $\displaystyle{G_{k}:=\liminf_{n\to\infty}p_{n+k}-p_{n}}$ should be approximately equal to $k(1+H_{k})$, where $H_{k}$ is the $k$-th harmonic ...
0
votes
0answers
67 views

Factoring a polynomial in a specific manner

Let $f(x,y,z) = ax^d + b y^k z^{d-k} + c$, where $a,b,c \in \mathbb{C}$ and $abc \ne 0$, and $d \geq 3$. Let $d'$ denote the smallest non-negative integer $l$ such that there does not exist a positive ...
3
votes
2answers
224 views

Residue for the generating function of the Euler totient function

Let $\varphi$ be the Euler totient function, and let us define the function $f(z)$ by the series $$ f(z) := \sum_{n=1}^{\infty} \varphi(n) z^n $$ Since $0\le \varphi(n)\le n$, I believe this gives a ...
-3
votes
0answers
58 views

prove that the sequence a_n = [n((2)^(1/2))] + [n((3)^(1/2))] contains infinitely many even and odd integers [closed]

im not sure how to proceed.i have done a few problems similar to this but i can't solve this one. thanks for any help.( by the way this is a problem from croatia team selection test )
11
votes
2answers
460 views

For what real $t$ is $\{n^t : n \geq 1\}$ linearly independent over $\mathbb{Q}$?

It's straightforward that $t$ must be irrational. I have googled many variations of this question and browsed through some books on transcendental number theory. There is much that is said about when ...
1
vote
0answers
45 views

Runs of consecutive numbers all of which are rebel numbers [migrated]

A positive integer is said to be a rebel number if it is the product of numbers none of which share any of the original number´s digits. Thus 10 = 2 x 5 is a rebel number, while none of the primes is. ...
83
votes
7answers
3k views

Is the set $ AA+A $ always at least as large as $ A+A $?

Let $A$ be a finite set of real numbers. Is it always the case that $|AA+A| \geq |A+A|$? My first instinct is that this is obviously true, and there is a one-line proof which I am foolishly ...
4
votes
0answers
125 views

Lower bound on class number of binary quadratic forms of discriminant of the form $n^2+4$

While searching for a use for the "sum invariant" of indefinite binary quadratic forms of discriminant $D = n^2 + 4$ (see https://cs.uwaterloo.ca/journals/JIS/VOL17/Smith/smith5.html), I believe I ...
12
votes
4answers
696 views

Random Diophantine polynomials: Percent solvable?

Suppose one generates a random polynomial of degree $d$ with integer coefficients uniformly distributed within $[-c_\max,c_\max]$. For example, for $d=8$, $|c_\max|=100$, here is one random ...
1
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0answers
119 views

All non-split Cartan subroups of $GL_2(\mathbb{Z}/n\mathbb{Z})$ are conjugate

Let $n>1$ be a positive integer and let $R$ be an order in an imaginary quadratic field with discriminant prime to $n$. Let $A=R/nR$ and let $\lbrace 1, \alpha \rbrace$ be a ...
11
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0answers
303 views

Decidability of $x^3+y^3+z^3 = c$

I wondering if it is known whether the following problem is algorithmically decidable or undecidable by Turing machines: given an integer c, determine if there are integers $(x,y,z)$ such that ...
3
votes
0answers
280 views

Do those manifolds atrached to L-functions give rise naturally to motives? [closed]

Edited after Will Sawin's comment: Consider the set $\mathcal{M}$ of all automorphic L-functions belonging to the Selberg class. Such a set is closed for the product $.$ and the tensor product ...
11
votes
1answer
402 views

Can Shor's Algorithm be modified to run efficiently on a classical computer?

Shor's algorithm is an algorithm which factors integers in polynomial time on a quantum computer. If one tries to run it on a classical computer, one runs into the problem that the state vector that ...
10
votes
0answers
399 views

Why are solutions to $\sqrt[k]{x_1^k+x_2^k+x_3^k+x_4^k}$ for $k=2,3$ curiously smooth?

Given an integer solution $s_m$ to the system, $$x_1^2+x_2^2+\dots+x_n^2 = y^2$$ $$x_1^3+x_2^3+\dots+x_n^3 = z^3$$ and define the function, $$F(s_m) = x_1+x_2+\dots+x_n$$ For $n\geq3$, using an ...