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

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Solving system of linear diophantine equations 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 ...
user1868607's user avatar
-4 votes
0 answers
66 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\...
Sylvain JULIEN's user avatar
0 votes
0 answers
18 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 ...
opfromthestart's user avatar
14 votes
0 answers
179 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 $...
Terry Tao's user avatar
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-6 votes
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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 ...
user524928's user avatar
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42 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\...
Notamathematician's user avatar
5 votes
2 answers
216 views

Prime differences and zero multiplicity

Paul Erdős conjectured, for consecutive primes, 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), Selberg showed that a ...
Felixson's user avatar
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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 ...
Mikhail Borovoi's user avatar
4 votes
1 answer
254 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 ...
Nandor's user avatar
  • 289
2 votes
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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 : $$u_0 = x$$ $$ \mathrm{If}, u_n \le c, \mathrm{then}, u_{n + 1} = u_n + (1 - c) $$...
Azoth's user avatar
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2 votes
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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} ...
Daniel Lang's user avatar
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, ...
John Rached's user avatar
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 ...
Yuri Bilu's user avatar
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1 vote
1 answer
146 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 $\...
Nobody's user avatar
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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 ...
kindasorta's user avatar
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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, ...
XL _At_Here_There's user avatar
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 ...
Simon Pohmann's user avatar
3 votes
0 answers
332 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?
Jon23's user avatar
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-1 votes
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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 ...
Bubbler's user avatar
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1 vote
0 answers
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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<...
user2284570's user avatar
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 ...
MAS's user avatar
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9 votes
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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 $\...
Taras Banakh's user avatar
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3 votes
2 answers
500 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 ...
Guruprasad's user avatar
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 ...
user2284570's user avatar
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 (...
H A Helfgott's user avatar
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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\...
Dumbest person on earth's user avatar
6 votes
0 answers
258 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 ...
Jorge Zuniga's user avatar
  • 2,202
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 ...
user15243's user avatar
  • 474
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\...
Yaakov Baruch's user avatar
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 ...
Bogdan Grechuk's user avatar
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 ...
HardProblemHero's user avatar
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 $...
Roland Bacher's user avatar
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 ...
psubodiosa's user avatar
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 ...
T. Amdeberhan's user avatar
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 ...
Marcos Cramer's user avatar
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?
Manfred Mueller-Spaeth's user avatar
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 ...
Ingrid's user avatar
  • 23
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 ...
BanAna's user avatar
  • 61
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 ...
Olivier's user avatar
  • 10.2k
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_{...
Hhhhhhhhhhh's user avatar
  • 1,032
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 $n \geq m$. ...
Hetong Xu's user avatar
  • 579
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 ...
Maximilian Janisch's user avatar
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 $...
Noname's user avatar
  • 191
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 ...
Joshua Stucky's user avatar
13 votes
2 answers
976 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 ...
Kyaw Shin Thant's user avatar
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 ...
Bogdan Grechuk's user avatar
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 $(r,s)$...
Henri Cohen's user avatar
  • 11.4k
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 ...
Nandakumar R's user avatar
  • 5,401
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-...
kindasorta's user avatar
  • 1,373
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 ...
Nobody's user avatar
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