Questions tagged [nt.number-theory]
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
10,877 questions
3
votes
0answers
46 views
Conjectured count of monogenic rings of fixed rank
By a monogenic ring of rank $n$ we mean a ring $R = \mathbb{Z}[\theta]$, where $\theta$ is an algebraic integer of degree $n$ over $\mathbb{Q}$. Put $f_\theta(x)$ for the minimal polynomial of $\theta$...
2
votes
0answers
193 views
Does multiplication increase entropy?
Does multiplication increase entropy?
The Shannon entropy of a number $k$ in binary digits is defined as
$$ H = -\log(\frac{a}{l})\cdot\frac{a}{l} - \log(1-\frac{a}{l})\cdot (1-\frac{a}{l})$$
where $...
-1
votes
0answers
33 views
Solving a promise bivariate modular polynomial system
Consider a prime $p$ and a system of $t$ equations of form
$$\alpha_ix^{a_i}y^{b_i}+\beta_ix^{c_i}y^{d_i}\equiv\gamma_i\bmod p$$
where $\alpha_i,\beta_i,\gamma_i\in\mathbb Z$ are chosen in $[0,p-1]$ ...
-2
votes
0answers
25 views
Probability of count of solutions of a random bivariate modular system of particular form
Consider a prime $p$ and a system of $t$ equations of form
$$\alpha_ix^{a_i}y^{b_i}+\beta_ix^{c_i}y^{d_i}\equiv\gamma_i\bmod p$$
where $\alpha_i,\beta_i,\gamma_i\in\mathbb Z$ are uniformly chosen in $[...
3
votes
0answers
77 views
Which Shimura varieties admit or don't admit $p$-adic uniformization by Drinfeld spaces?
$p$-adic uniformization is a powerful tool for studying Shimura curves and Shimura varieties. For instance, we know cohomology groups of Drinfeld spaces, so we know some information about the Shimura ...
3
votes
0answers
91 views
On a set of natural numbers
For any natural number $x$, let $f(x)$ be the natural number whose representation in base 3 is the same as the binary representation of $x$ (for instance, $f(5)=10$ because $5=101_2$ and $101_3=10$). ...
5
votes
0answers
153 views
Factorizations as a product of primes minus one
Let $x$ be a positive rational number. I am interested in factorizing $x$ as a product of primes minus one. In fact, I would also like make sure the primes in the decomposition are distinct, and I ...
3
votes
1answer
234 views
Isomorphism of the $\ell$-adic Tate module of an elliptic curve with CM
Let $E$ be an elliptic curve over $K$ (totally real number field) with complex multiplication by the field $L$. Let $\psi$ be the Grössencharacter associated to $E$, assume that $\psi$ of type $(-r,0)$...
13
votes
1answer
229 views
Are the logarithms of the integer polynomials discrete in $L^1$ of the unit circle?
Tautologically, the integer polynomials form a discrete set in $L^1$ of the unit circle. On the other hand, a set of logarithms ordered by norm becomes generally rather denser than the original set.
...
-2
votes
0answers
87 views
Modifying Liouville Number Construction
Liouville numbers are transcendental numbers that can be well approximated by rational numbers. A number x is a Liouville number if for every natural number $n$ there exist infinitely many pairs of ...
6
votes
0answers
136 views
+50
Existence of radial limits of products of certain power series and $1-x$
Let $V$ be an arbitrary set of infinitely many positive integers, and let: $$\varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}z^{v}$$ Let $T_{V}$ denote the set of all $t\in\left[0,1\...
3
votes
1answer
114 views
Bounds on Artin conductors over function fields
Let $L/K$ be a geometric Galois extension of function fields over $\mathbb F_q$. Let $\chi$ be a non-trivial irreducible character of $\text{Gal}(L/K)$. According to Michael Rosen's Number Theory in ...
3
votes
0answers
348 views
Four-square Conjecture
Lagrange's four-square theorem states that every nonnegative integer
can be written as the sum of four squares. My following conjecture is much stronger than this classical theorem.
Four-square ...
7
votes
0answers
165 views
Does every multiplicative function have a logarithmic average?
Is it true that for every completely multiplicative function $f:\mathbb N\to\mathbb C$
with $|f(n)|=1$ for all $n$, the logarithmic average
$$\lim_{N\to\infty}\frac1{\log N}\sum_{n=1}^N\frac1nf(n)\...
5
votes
0answers
85 views
Asymptotic expansion for the average of $\omega(n)^2$
Let $\omega(n)$ be the prime factors counting function. I computed that for any $k\geq 0$, there exist certain constants $c_{-1},c_0,c_1,c_2,...c_k$ such that
$$\sum_{n\leq x}\omega(n)^2=x(\log\log x)...
2
votes
1answer
166 views
Relation between arithmetic functions and modular forms
Generating functions of many multiplicative arithmetic functions of longstanding interest (e.g., sum-of-divisors function, number-of-partitions function) turn out to be Fourier expansions of modular ...
0
votes
0answers
129 views
Definition of an invariant differential of an elliptic curve
I am somewhat confused by the definition of the invariant differentials in J. Silverman's book The Arithmetic of Elliptic Curves.
Let $E$ be an elliptic curve with Weierstrass equation $F(x,y)=0$. ...
2
votes
0answers
51 views
Density of integral values of a rational function
Let $\mathbf{x} = (x_1, \cdots, x_n)$, and consider a rational function $F : \mathbb{R}^n \rightarrow \mathbb{R}$ be given by
$$\displaystyle F(\mathbf{x}) = \sum_{i = 1}^m \frac{Q_i(x_1, \cdots, x_{...
-1
votes
0answers
85 views
$12n+5=x^2+y^2+z^2$ with $x\in\{2^a5^b:\ a=1,2,3,\ldots\ \text{and}\ b=0,1,2,\ldots\}$
By the Gauss-Legendre theorem on sums of three squares, for each $m\in\mathbb N=\{0,1,2,\ldots\}$, we may write $4m+1$ as the sum of three squares. Surprisingly, I found that this classical result ...
0
votes
2answers
152 views
All the integer solutions of a certain semi-algebraic system
I would like to find all integer solutions of the following system:
$$a+b+c+ab+ac+bc=-2,$$
$$a,b,c\le a+b+c-1.$$
One solution is $2,2,-2$. Is it possible to describe all others?
11
votes
0answers
92 views
Hypergeometric representation of Eisenstein series
It is well known (Fricke ?) that $E_4^{1/4}$ and $E_6^{1/6}$ can be represented as Gauss hypergeometric functions of $1728/j$ and $1728/(1728-j)$ respectively.
The same result is true in levels $2$, $...
3
votes
0answers
148 views
For any finite subset $A \subset \mathbb{R}$ we have that $\left| \frac{A+A}{A+A}\right| \gg |A|^2 $
I am trying to understand how sumset theory is actually used in other parts of math or within additive combinatorics. Here are some results I have found in this paper from 2018 ([1], [2]):
Thm (...
2
votes
0answers
105 views
Distribution of Goldbach's weak-conjecture's prime-triples
From Harald Helfgott's proof of Goldbach's weak conjecture,1
we know that every odd number $> 7$ is the sum of three odd primes.
If $n$ is such an odd number, say that two sums that yield $n$
are ...
9
votes
2answers
548 views
Number of solutions mod p and Betti numbers
Suppose $X$ is proper flat scheme over Sepc$\mathbb{Z}$. If $p$ is a large prime, then by Weil conjectures one can recover the Betti numbers of $X$ from the size of $X(\mathbb{F}_{p^n})$ for all $n$. ...
1
vote
1answer
125 views
Local heights in Vojta's conjecture
I am a complex geometer trying to parse Vojta's conjecture on rational points, and I have a very basic misunderstanding (I apologize if this is too easy for MO).
Let $X$ be a variety over a number ...
2
votes
1answer
165 views
Choosing finite subsets of natural numbers
Let $t>0$ and $\delta\in\big(0,\frac12\big)$ be fixed. For any $k\in\mathbb{N}$ let $I_k,J_k\in\mathbb{N}$ be finite subsets of natural numbers with cardinalities denoted as $|I_k|,|J_k|$, ...
6
votes
1answer
765 views
Does Chowla's conjecture on the Liouville function imply the Riemann hypothesis?
A paper see here on arXiv claims that Chowla's conjecture (applied to the Liouville function instead of the Mobius function), i.e., that
$$
\lim_{N\rightarrow \infty} \sum_{n\leq N}
\lambda(n+a_1) \...
1
vote
0answers
196 views
A question about a set of prime numbers
Let $n$ an integer sufficiently large.
I'm looking for a "big" set $S$ of prime number $p$ satisfying $10.n \geq p > \sqrt{18n} $ and $p| C_{9.n}^{k-n}C_{8n+k}^{8.n}$ for all integer $k$ ...
9
votes
0answers
260 views
Rational points of a “famous” elliptic curve
The following problem has already been discussed in mathoverflow, for example here
It is essentially an elliptic curve of rank $1$, a generator of the Mordell-Weil group being the point $Q = (\frac{4}...
4
votes
0answers
120 views
Limit of the real part of a geometric sequence
I came across the following problem, which turned out to be surprisingly hard:
Show that $\underset{n\rightarrow \infty}{\lim} \left| \mathrm{Re}((\frac{1+i\sqrt{7}}{2})^n)\right| = \infty.$
...
5
votes
1answer
670 views
Extending prime numbers digit by digit while retaining primality [closed]
I looked at a table of primes and observed the following:
If we choose $7$ can we concatenate one digit to the left so as to form a new prime number? Yes, concatenate $1$ to obtain $17$. Can we do ...
2
votes
1answer
104 views
Euler characteristics in the rank one case
Suppose $E$ is an elliptic curve over a number field with good ordinary reduction at the primes above a fixed odd prime $p$. We are interested in the Iwasawa theory over the cyclotomic $\mathbb{Z}_p$ ...
2
votes
0answers
95 views
Queries on distribution of prime divisors by magnitude?
Erdos-Kac law state a typical number of magnitude $n$ has $\log\log n$ prime factors and we know probability of square free integers is $\frac{6}{\pi^2}$.
What is the probability distribution of ...
3
votes
0answers
90 views
Magnitude and distribution of largest prime factor?
Erdos-Kac law state a typical number of magnitude $n$ has $\log\log n$ prime factors.
What is magnitude and distribution of largest prime factor of typical magnitude $n$ natural number?
What ...
1
vote
1answer
56 views
Explicit bivariate quadratic polynomials where Coppersmith is better than standard solver?
http://www.numbertheory.org/pdfs/general_quadratic_solution.pdf gives a general method to solve quadratic bivariate diophantine equation while Coppersmith introduced a method to solve bivariate ...
8
votes
0answers
126 views
Strong uniqueness of Euler's totient function
Let $f:\mathbb N\to \mathbb C$ be some arithmetical function. Define $\varphi_f(n)$ by the following formula:
$$
\varphi_f(n)=\sum_{\substack{k\leq n \\ (k,n)=1}}f(k).
$$
In other words, $\varphi_f(...
1
vote
1answer
188 views
Representing integers efficiently with quadratic polynomials
For every large enough $T$ given four integers $a,b,c,d$ with absolute value less than $T^2$ are there integers $w_1,x_1,y_1,z_1,w_2,x_2,y_2,z_2$ with absolute value less than $T$ such that
$$w_1x_1+...
1
vote
0answers
91 views
Lower and upper bounds of the distance between two Frobenius numbers
I consider two sequences of numbers: $A=\{a_1,...,a_{m-1},n\}$ and $B=\{n-a_{m-1},...,n-a_1,n\}$, where $a_1 < a_2 < ... < a_{m-1} < n$ and $\gcd(A) = \gcd(B) = 1$.
I investigate the ...
17
votes
2answers
656 views
$P(x)=P(y)$ has infinitely many integer solutions
Determine all polynomials $P(x)$ with integer coefficients such that $P(x)=P(y)$ has infinitely many integer solutions in integer $x$ and $y$ with $x \neq y$.
Choose $P(x)=a_n(x-k)^{2n}+a_{n-1}(x-k)^{...
1
vote
0answers
467 views
Are either $\pi + e$ or $\pi e$ transcendental if we add or multiply digit-wise? [closed]
Since $x^2 - (e + \pi)x + e \pi = (x - \pi)(x - e)$ has transcendental roots, we know that the coefficients are not both rational, and not even algebraic (see comment by José).
My question is, can we ...
3
votes
0answers
69 views
Polynomial equations parametrized by binary forms
Consider the equation
$$\displaystyle Ax^p + By^q = Cz^r, A,B,C \in \mathbb{Z}, \gcd(x,y,z) = 1, p,q,r \geq 2.$$
When $p^{-1} + q^{-1} + r^{-1} > 1$, the above equation is called spherical and ...
2
votes
1answer
72 views
Representation of a finite group over a finite field from rational representations
Suppose $G$ is a the cyclic group of $n$ elements and $p$ is a prime not in $n$. $G$ has an action on the cyclotomic field as a $\mathbb{Q}$-vector space $\mathbb{Q}[X] / \langle \Phi_n(X) \rangle$ by ...
3
votes
4answers
244 views
Approximately satisfying simultaneous vector linear diophantine equations?
Pick three $a,b,c$ vectors in $\mathbb Z^n$ uniformly with $\max(\|a\|_\infty,\|b\|_\infty)<T$ and $\|c\|_\infty<T^2$ and an $\epsilon>0$.
Assume $a$ and $b$ are coordinatewise coprime (...
16
votes
0answers
414 views
Shimura varieties and connected components
Let $G$ be a connected reductive algebraic group over $\mathbf{Q}$. I've seen two slightly different definitions in the literature of the Shimura variety of level $U$, for $U \subseteq G(\mathbf{A}_{\...
2
votes
0answers
150 views
Conjectured primality test for numbers of the form $N=4 \cdot 3^n-1$
This is a repost of this question.
Can you provide proof or counterexample for the claim given below?
Inspired by Lucas-Lehmer primality test I have formulated the following claim:
Let $P_m(x)=2^{...
0
votes
1answer
66 views
The maximum difference between the number of elements in the two sets of equal length of consecutive numbers that divisible by some prime numbers
Suppose that $p_1,...,p_k$ are distinct prime numbers.
Let $f(n,l)$ be equal to the number of elements from set $\{n+1,n+2,...,n+l\}$ that are divisible by some $p_1,...,p_k$. Is it true that
$$\...
4
votes
0answers
100 views
Is there a converse to Vatsal's theorem on congruence of p-adic L-functions?
Let $f=\sum_n a_n(f) q^n$ and $g=\sum_n a_n(g) q^n$ be normalized (cuspidal) newforms whose Fourier coefficients are contained in the p-adic field K for which the uniformizer of $\mathcal{O}_K$ is ...
0
votes
0answers
99 views
Why can there be holomorphic modular forms of negative half integral weight?
In Shimura's paper "ON THE HOLOMORPHY OF CERTAIN DIRICHLET SERIES", he constructed a family of Eisenstein series $E(z,s)$ by summing factor of automorphy. $E(z,s)$ is of negative half-integral weight, ...
0
votes
0answers
91 views
Prime counting function estimate sieve of Eratosthenes-Legendre
I'm trying to arrive at estimate 1.17 (page 21) of Koukoulopoulos lecture notes [https://dms.umontreal.ca/~koukoulo/documents/notes/sievemethods.pdf]
$$\#\{n \leq x : p|n \Rightarrow p > \sqrt{x}\}...
2
votes
0answers
79 views
Existence of nontrivial finite sub-modules in the cyclotomic extension
It is a well established fact (by Greenberg) that if $p$ is a prime of good ordinary reduction of an elliptic curve $E/\mathbb{Q}$, then the dual of the Selmer group, denoted by $X(E/\mathbb{Q}_{cyc})$...
