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
15,877
questions
0
votes
0
answers
47
views
Solving system of linear diophantine equations with exponential coefficients over the integers
In general, solving a system of linear diophantine equations over the integers is polynomial time solvable on the size of the coefficients of the equations.
I am interested in an extension of this ...
-4
votes
0
answers
71
views
Goldbach's conjecture with Claude 3 [closed]
I wanted to test the mathematical abilities of Claude 3, so I typed a prompt leading to the following conjecture:
Assume Goldbach's conjecture and denote by $r_{0}(n):=\inf\{r\geq 0\mid(n-r,n+r)\in\...
0
votes
0
answers
20
views
Find transseries from difference equation
I want to find a method to solve equations of the form
$f(x+1)=f(x)+g(x)$ for a given function $g$ and $f(x)=0$.
The paper here has solutions for $f(x+1)=\lambda(x)f(x)+g(x)$, which is more general ...
14
votes
0
answers
202
views
Proofs of the valence formula that avoid tricky contours?
$\DeclareMathOperator\ord{ord}\DeclareMathOperator\Im{Im}$The valence formula for a modular form asserts that if $f: \mathbf{H} \to \mathbf{C}$ is a modular form of weight $k$ on the upper half-plane $...
-6
votes
0
answers
40
views
Is the Square-free Test A Polynomial Time Algorithm?
The paper
"MILLER’s PRIMALITY TEST",
Volume 8, number 2 INFORMATION PROCESSING LETTERS February 1979
by H.W. LENSTRA, Jr.,
claims to have a square-free algorithm of polynomial time ...
0
votes
0
answers
44
views
On partitions into distinct parts and binary
Let $a(n)$ be A000009 (i.e., number of partitions of $n$ into distinct parts or number of partitions of $n$ into odd parts).
Let
$$
b(n) = \sum\limits_{i=0}^{n} a(i)
$$
Let
$$
\ell(n) = \left\lfloor\...
6
votes
2
answers
272
views
Prime differences and zero multiplicity
Concerning gaps between consecutive primes, Paul Erdős conjectured that:
$$\sum_{p_n < x} (p_n -p_{n-1})^2 = O(x \log x)$$
Let's call this hypothesis EH. Assuming the Riemann hypothesis (RH), ...
2
votes
0
answers
54
views
Estimating the cardinality of the set of conjugacy classes of subgroups in a finite group of given order
1. Let $G$ be a finite group of order $n$. I need an estimate for the number $c$ of conjugacy classes of subgroups $D\subseteq G$.
Note that any subgroup of $G$ contains $1_G$, and so the set of all ...
4
votes
1
answer
259
views
Subgroup of p-adic units
Let $\smash{\widehat{\mathbb Z}}^\times=\prod_p{\mathbb Z}_p^\times$
be the unit group of the ring $\widehat{\mathbb{Z}}$, which is the profinite completion of $\mathbb Z$.
We give it the product ...
2
votes
0
answers
103
views
A sequence linked to irrationality
Let $0 < c < 1$ be a real number and $ x \in \mathbb{R}$. We define the sequence $(u_n)_{n \in \mathbb{N}}$ by :
$$u_0 = x$$
$$ \mathrm{If}, u_n \le c, \mathrm{then}, u_{n + 1} = u_n + (1 - c) $$...
2
votes
0
answers
96
views
Gaussian primes in translations of lattices in $\mathbb{Z}[i]$
I am considering undertaking some independent research in my summer break studying Gaussian primes in translations of lattices in $\mathbb{Z}[i]$, i.e. sets of the form $ \{ a+sx+tw:s,t \in \mathbb{Z} ...
2
votes
0
answers
99
views
Bourgain-Gamburd-like theorems in the non-algebraic case
For $\mu$ a Borel probability measure on the compact group $G=\operatorname{SU}(d)$, Bourgain-Gamburd prove that the spectral radius of the associated operator on $L^2(G)$ is strictly less than one, ...
7
votes
1
answer
498
views
Original proof of Hilbert irreducibility theorem
Does there exist a modern exposition of Hilbert's original (1892) proof of the Hilbert irreducibility theorem? Of course, I can (and will) read Hilbert's original article, but I would feel more ...
1
vote
1
answer
149
views
Existence of odd mod $p$ Galois representations whose image is $p'$-group
Let $K$ be a number field and let $G_K$ be the absolute Galois group of $K$. Let $p$ be an odd prime and $\mathbb{F}_p$ be a finite field of order $p$. Can we always find a continuous representation $\...
1
vote
0
answers
97
views
Multiplicities of Galois representations in the semisimplification of the reduction of a Tate module
Let $C$ be a smooth proper curve, of genus $g$ over a number field $K$. Let $v$ be a prime of good reduction for $C$ above $p>2$, and let $T_pJ$ denote the $p$-adic Tate module of $J$, the Jacobian ...
0
votes
0
answers
126
views
A question and reference about Bombieri's article continued fraction of algebraic numbers
Above the Comments in the article continued fraction of algebraic numbers, there are some words on the unboundedness/cycle of coefficients of continued fraction of algebraic numbers "Thus, ...
1
vote
0
answers
57
views
When is the number-theoretic transform of small vectors again small?
I am currently working on an idea in the context of lattice-based cryptography, but the problem that I am currently stuck on seems to have almost nothing to do with lattices anymore.
In particular, my ...
3
votes
0
answers
333
views
Analytic number theory and condensed mathematics
As of 2024, are there current or planned applications of condensed mathematics to analytic number theory? If so, what are suggested readings?
-1
votes
0
answers
62
views
On the full list of near-repdigit perfect powers
I'm interested in the full list of perfect powers ($a^b$ where $a, b \in \mathbb{Z}$, $a \ge 1$, $b \ge 2$) that are near-repdigit in base 10. A near-repdigit is a $k$-digit number where $k \ge 2$ and ...
1
vote
0
answers
50
views
How to check that a number probably/likely has a divisor having a specific bit length/in range?
Given a randomly generated $\alpha\in\Bbb N$ where $\alpha$ is large thus hard to factor (no small prime composites). How to check that a divisor $F\in\Bbb N$ with a specific bitlengh $n\in\Bbb N∧n<...
2
votes
0
answers
76
views
Is the extension field by zeros of $x^{2m}-p^{2(m-1)}=0$ over $\mathbb Q_p$ totally ramified?
Let $m \geq 2$ be an integer. Consider the polynomial $f(x)=x^{2m}-p^{2(m-1)} \in \mathbb{Q}_p[x]$.
I want to study the field extension by the zeros of $f(x)$ over $\mathbb{Q}_p$.
What is the degree ...
9
votes
0
answers
145
views
Who was the first to prove that the automorphism group of a finite field is cyclic and is generated by the Frobenius automorphism?
$\DeclareMathOperator\Aut{Aut}$It is well-known that the automorphism group $\Aut(F)$ of a finite field $F$ of characteristic $p$ is cyclic of order $n$ where $|F|=p^n$. Moreover, the cyclic group $\...
2
votes
2
answers
506
views
Sum of three square is a square and sum of their product taken two at a time is also a square
Let $a^2 + b^2 + c^2 = X^2$ and
$$(ab)^2 + (ac)^2 + (bc)^2 = Y^2$$
Such that $a,b,c,x,y$ are all Integers
How to find All non trivial solutions ?
Is there any parametrization which gives many ...
1
vote
0
answers
86
views
How to know if a random natural number is a probable semiprime?
Let that $n\in\Bbb N$ generated from a hash function where $n$ is long enough to be hard to factor in the gnfs algorithm. How to check if $n$ is probably a semi‑prime in a faster way than factoring it ...
3
votes
0
answers
176
views
Do all polynomials (other than generalized cyclotomic polynomials) have the spaced polynomial property?
Anna Erschler just asked me a question that is posed as Question 1.2 in her recent preprint with J. Frisch and M. Rychnovsky. I am asking it here with her permission - since I find it interesting (...
1
vote
0
answers
51
views
A question on generalized bases
I just came to know that it is possible to define a generalized base as an infinite sequence of natural numbers $\mathbf b=(b_1,b_2,\dots)$ where $b_i\ge 2$ for all $i$. With this definition, any $m\...
6
votes
0
answers
260
views
+100
How to prove these identities for $\log(2)$ based on $_3F_2$ integrals?
In this MO post I have placed 4 Ramanujan-type hypergeometric series found using the LLL algorithm for fast computing of some logarithms. I could prove 3 of them by means of classical methods based on ...
1
vote
0
answers
112
views
A question related to Kirillov model
I am reading Jackson - The theory of admissible representations of $\operatorname{GL}(2, F)$ and am not able to understand the following map related to Kirillov model. This result appears on p. 54:
I ...
17
votes
1
answer
3k
views
Assuming the Collatz conjecture is false, what is known about the size of the false set?
If the Collatz conjecture is strongly false, in the sense that there is an infinite orbit, let $S_n$ be the set of natural numbers $\le n$ whose orbit goes to infinity.
If $c=\liminf _{n\rightarrow\...
6
votes
0
answers
225
views
$1 + 3 x^3 + x y^2 + 6 y z^2 = 0$ - the new shortest open cubic equation
Are there integers $x,y,z$ such that
$$
1 + 3 x^3 + x y^2 + 6 y z^2 = 0 \,\, ? \quad\quad (1)
$$
If the length of an equation is the sum of degrees of monomials plus sum of logarithms of the ...
1
vote
1
answer
83
views
Number of solutions for linear modular equations given GCD
We are currently investigating a problem involving number theory, an area outside our field of expertise.
Let $n$ be a positive integer. Consider two pairs of integers $(j,k)$ and $(j′,k′)$ as ...
9
votes
1
answer
397
views
Jacobi symbols for two-square sums of primes
Given a prime $p\equiv 1\pmod 4$, Fermat's two-squares theorem discovered by Girard
states that there exists two integers $A,B$ such that
$p=A^2+B^2$.
For all primes up to $10^7$ the integers $A$ and $...
5
votes
1
answer
350
views
Lindelöf hypotheses for derivatives of zeta
The Lindelöf hypothesis says that if we have:
$$\zeta(\sigma+iT)=\mathcal O(T^a)$$
Then if one considers $\sigma=1/2$ then $\inf a=0$. Further, from convexity and the functional equation this implies ...
2
votes
1
answer
271
views
Counting points on elliptic curves
Consider the Legendre family of elliptic curves
$$E_a: y^2=x(x-1)(x-a).$$
Let $p$ be an odd prime.
QUESTION. Is the following true? If $p\equiv 3\pmod4$ then number of solutions to $E_2$
over the ...
5
votes
2
answers
514
views
Representing natural numbers as sums of distinct prime powers
I am investigating whether every natural number $n > 18$ can be represented as a sum $p_1^{m_1} + \dots + p_k^{m_k}$, where $p_1, \dots, p_k$ are distinct primes, and $m_1, \dots, m_k$ are distinct ...
3
votes
1
answer
409
views
Is $n!$ divisible by $(n + 1)(n + 2)\cdots(n + d)$?
Is $n!$ divisible by $(n+1)(n+2)\cdots(n+d)$ when $n\ge 7$ is prime and $n+1,n+2,\ldots,n+d$ form a prime gap?
2
votes
1
answer
86
views
Stabilizing conjugacy classes of integer matrices
$\DeclareMathOperator{\Conj}{Conj} \DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\id}{id} \newcommand\Z{\mathbb{Z}}$
For an $n \times n$ integer matrix $A \in \GL_n(\Z)$, let $\Conj(A)$
be the ...
6
votes
1
answer
200
views
What is the difference between Hida and Coleman families?
From my understanding: Hida families and Coleman families of modular forms are roughly given by $p$-adic modular forms whose $q$-expansion at classical weights is "close" to a $q$-expansion ...
8
votes
1
answer
186
views
Is there something I am missing about the computation of the $p$-part of the class groups of cyclotomic fields?
Well, the answer of the question in the title in certainly Yes, many things in fact, but let me be more precise.
In 1958, Serre gave a Bourbaki talk on the recent works of Iwasawa on class groups in ...
0
votes
0
answers
52
views
Average of number of divisors of shifted exponential sequence
Let $a$ be a fixed positive integer greater than 1. We define the sequence $u_n=a^{n}-1$ for all positive integers $n$. Then are there any results in literature for asymptotic value of the sum $$\sum_{...
2
votes
1
answer
75
views
Reference Request: Possible generalizations of the stability of $\gamma$-factors
$\DeclareMathOperator\GL{GL}$
Let $F$ be a nonarchimedean local field. Suppose $\pi, \sigma$ are irreducible admissible representations of $\GL_{n}(F)$ and $\GL_{m}(F)$ respectively, with $n \geq m$. ...
2
votes
0
answers
162
views
Asymptotics of $\vartheta(x+y)-\vartheta(x)$, where $\vartheta$ is the Chebyshev function, when $y\in[x^\alpha, x]$ for some $\alpha\in(0,1)$
Introduction
Consider $\vartheta$ to be the Chebyshev function, that is, $\vartheta(x)$ denotes, for $x\in\mathbb N$, the sum $\sum_{p\le x,\ p\text{ prime}} \ln p$.
I am interested in asymptotics for ...
8
votes
2
answers
517
views
Do there exist positive integers $m$, $n$, $p$, $q$ such that $m>1$, $p\neq q$, $p$ and $q$ divide $mn^2 - 1$, and $mn$ divides $p - q$?
Do there exist positive integers $m$, $n$, $p$, $q$ such that $m>1$, $p\neq q$, $p$ and $q$ divide $mn^2 - 1$, and $mn$ divides $p - q$?
It seems numerically up to $n \leq 10^6$ that for $m=3$ or $...
6
votes
1
answer
215
views
Question about Größencharaktere in imaginary quadratic number fields
Presumably, one could ask this question for a Größencharakter in an arbitrary number field, but I'll restrict my attention to the case I'm interested in. Let $K$ be an imaginary quadratic field with ...
13
votes
2
answers
977
views
Using the Eichler-Selberg Trace formula to compute class numbers?
The Eichler-Selberg trace formula (Theorem 2.2 here) gives a relation between the trace of a Hecke operator acting on the space of cusp forms and sums of weighted class numbers of imaginary quadratic ...
29
votes
1
answer
1k
views
Can $9xy$ divide $1+x^2+x^3+y^2$?
Can $9xy$ divide $1+x^2+x^3+y^2$ for integers $x,y$? Equivalently, do there exist integers $x,y,z$ such that
$$
1 + x^2 + x^3 + y^2 + 9 x y z = 0 \quad ?
$$
This equation arises in the search for the ...
7
votes
0
answers
114
views
Finding a rational point of large height on an elliptic curve knowing a real approximation
Let $y^2=x(x^2+n)$ be an elliptic curve with $n\in\Bbb Z$ (the same question can of course
be asked for a general e.c). I know (e.g. it has rank 1) that there exists a nontrivial
rational point $(r,s)$...
0
votes
0
answers
69
views
Heronian tetrahedra with pairwise non-congruent, equal area faces
Reference: https://mathworld.wolfram.com/HeronianTetrahedron.html lists some Heronian tetrahedra that are disphenoids.
Are there any Heronian tetrahedra with all faces having same area but are ...
1
vote
0
answers
58
views
$F$-structure implies regular singularities + unipotent local monodromy?
Let $(\mathcal{E},\nabla)$ be a vector bundle with an integrable connection on a smooth quasi-projective $K$ scheme $X$, with $K$ a $p$-adic number field of characteristic $0$. Let $F$ denote a semi-...
3
votes
1
answer
184
views
Discrepancy between $\dim H^2(G, \mathrm{ad}(\bar \rho))$ and the number of relations in a minimal presentation of the universal deformation ring $R$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\ad{ad}\DeclareMathOperator\gen{gen}$Let $p$ be a prime and $G$ be a profinite group such that the pro-$p$ completion of every open subgroup is ...