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
4,691
questions with no upvoted or accepted answers
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Is this subset of a rigid space an admissible open?
Let $K$ be a $p$-adic field and let $X$ be the rigid space $ \operatorname{Max} K\langle T_1, T_2 \rangle$, i.e. the 2-dimensional closed unit ball.
Consider the sets $U := \{ |T_1| < 1\}$ and $V :...
- 36.2k
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89
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Is the ratio of a number to the variance of its divisors injective?
The variance $v_n$ of a natural number $n$ is defined as the variance of its divisors. There are distinct integer whose variances are equal e,g. $v_{691} = v_{817}$. However I observed that for $n \le ...
- 4,638
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226
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Diophantine applications of Paramodularity
I’ve asked this question to quite a few people in person and so far haven’t seen a good answer... but I believe one should exist, so here goes!
Ok, we all know how to (roughly) prove Fermat’s Last ...
- 562
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266
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Why does the longitude correspond to Frobenius in Arithmetic Topology, and other strange phenomena
I am trying to adress Morishita's book Knots and Primes to discover a bit about Arithmetic Topology, but some analogies puzzle me. I know that the correspondence should be addressed with a grain of ...
- 5,630
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204
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Complex isomorphism class of abelian varieties and $L$-functions
In his famous Mordell paper, Faltings proved that two abelian varietes $A_1, A_2$ defined over a number field $K$ are isogenous if and only if the local $L$-factors of $A_1, A_2$ are equal at every ...
- 25.5k
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231
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Counting elliptic curves by discriminant
Enumerating elliptic curves $E/\mathbb{Q}$ sorted by (the absolute value of) their minimal discriminants is a difficult open problem, as is the (likely easier) problem of counting elliptic curves $E/\...
- 25.5k
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135
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Complementary Bell numbers $B^{\pm}(24n+14)$
The complementary Bell numbers $B^{\pm}(n)$ are defined by the alternating sum of the Stirling numbers of the second kind, $S(n,k)$:
$$B^{\pm}(n)=\sum_{k=0}^n(-1)^kS(n,k),$$ and they count the ...
- 397
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457
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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 ...
- 591
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175
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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)...
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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 ...
user130124
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440
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Over what fields does the Mordell conjecture (Faltings's theorem) hold?
Inspired by this question, over what fields is the Mordel conjecture known to be true?
For instance, is it true over fields of finite type (that is, fields finitely generated over their prime ...
- 7,646
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281
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Symmetry of the distribution of prime gaps
Following Positive proportion of logarithmic gaps between consecutive primes let for given $\lambda$, $\alpha$ and for any $x$ all positive the quantities $S^{-}_{\lambda,\alpha}(x):=\#\{p_{n+1}\...
- 6,932
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Is there a prime $q$ smaller than a given prime $p>5$ such that the inverse of $q$ modulo $p$ is an integer square?
Let $p$ be a prime. For each $k=1,\ldots,p-1$ there is a unique $\bar k\in\{1,\ldots,p-1\}$ with $k\bar k\equiv1\pmod p$, and we call $\bar k$ the inverse of $k$ modulo $p$. In 2014 I investigated ...
- 14.5k
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Multivariable multilinear homogeneous polynomials with co-prime coefficients representing 1
Suppose $F(x_1,\dots,x_n)$ is a homogeneous polynomial in $n$ variables of degree $m$, which has degree $1$ in each of the variables. Suppose further that it has integer relatively prime coefficients. ...
- 255
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Is an algebraic number satisfying certain super-congruences a root of unity?
Let $K|\mathbb{Q}$ be a number field, $D$ its discriminant and $\mathcal{O}$ the ring of integers in $K$. Let $x\in K$ (or maybe $\in \mathcal{O}[\frac 1D]$) such that for all primes $p$ in $\mathbb{Q}...
- 171
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Exponential sums with prime power modulus
I am looking for an analogue of the following result of Fouvry and Katz for prime power modulus ("A general stratification theorem for exponential sums, and applications", J. reine angew. Math. 540 (...
- 25.5k
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190
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On the determinants $\det\left[(i\pm j)\left(\frac{i\pm j}p\right)\right]_{1\le i,j\le(p-1)/2}$
Let $p$ be an odd prime and define
$$D_p^+:=\det\left[(i+j)\left(\frac{i+j}p\right)\right]_{1\le i,j\le(p-1)/2}$$
and $$D_p^{-}:=\det\left[(i-j)\left(\frac{i-j}p\right)\right]_{1\le i,j\le(p-1)/2},$$
...
- 14.5k
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124
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Linear exponential sum with gcd
The sum $$\sum _{d,d'\leq D}\sum _{h,h'=1}^q(h,q)e\left (\frac {dh+d'h'}{q}\right )$$ is easily seen to be $$\ll q^{2+\epsilon }+D^2.$$ Indeed with a standard estimate for a linear exponential sum it ...
- 143
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Counting elliptic curves over a number field by their Faltings height
In this paper, Hortsch gives an asymptotic formula for the number of elliptic curves over $\mathbb{Q}$, given by their minimal Weierstrass model, of bounded Faltings' height.
In general, is it ...
- 25.5k
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241
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$\mu=0$ for CM Elliptic curves?
Let $E$ be an elliptic curve defined over $F$ with CM by $\mathcal{O}_K$ where $K$ is an imaginary quadratic field. We may assume the $F$ contains $K$ and also contains the $p$-division points, where $...
- 1,191
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Consecutive integers each of which has a large prime factor
There are many results about consecutive integers all having small prime factors. But what about consecutive integers each of which has a large prime factor?
More precisely, let $P(n)$ be the ...
- 341
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253
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Sums of two integer squares in arithmetic progressions
Is there an explicit formula in the literature for the number of representations of a positive integer $n$ as a sum of two integer squares, the second of which is divisible by $5$? So this means to ...
- 2,949
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From which layer we can apply Iwasawa's class number formula in Z_p extension?
Let $K$ be a number field. Let $K_\infty/K$ be a $\mathbb{Z}_p$ extension. Iwasawa proved that there are four integers $n_0,\mu,\lambda,\nu \geq 0$ such that for any $n\geq n_0$, $$\mathrm{ord}_p(...
- 781
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409
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Pisot conjugates
An informal version of my question is "If we have a Pisot number between 1 and 2 of a very large degree, is it true that all its other conjugates are very close to 1 in modulus?"
A more formal ...
- 2,085
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Is there some computational evidence of the $pq$ analog of Serre's conjecture?
The $pq$ analog of Serre's conjecture (see "Mod pq Galois representations and Serre's Conjecture"- Khare, Kiming) states that if $\bar{\rho}_1:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F}_p)$ is a ...
user130124
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Is there a conjectural analog of Ribet's theorem (Converse to Herbrand's Theorem) for Imaginary Quadratic fields?
For $p$ a prime, let $Cl(\mathbb{Q}(\mu_p))$ denote the class group of the extension of $\mathbb{Q}$ obtained by adjoining a primitive $p$th root of unity. Associated to an eigenform of weight 2 and ...
user130124
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108
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"middle" partial denominator in continued fraction expansion of square roots
Suppose $d$ is a positive integer that is not a perfect square such that the negative Pell equation, $x^{2}-dy^{2}=-1$ has no solution. Then we know the minimal period of the continued fraction ...
- 51
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Reference Request: Cohomology and limits of coherent topoi. (Non-abelian case)
SGA 4 VI Discusses finiteness conditions one can impose on topoi to make limits behave correctly. I am not that familiar with SGA but it is my impression that this expose only discusses abelian ...
- 415
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228
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Can we write each positive integer as $x^2+y^2+\varphi(z^2)$?
As odd squares are congruent to $1$ modulo $8$, any integer of the form $4^k(8m+7)$ with $k,m\in\mathbb N=\{0,1,2,\ldots\}$ cannot be written as the sum of three squares.
To avoid such congruence ...
- 14.5k
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337
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Lower bound for some sums of roots of unity
Let $n$ be a positive integer (assume $n$ is prime for simplicity), and let $x_k = \pm1$, for $k = 0,1,2,..., n-1$. Let $\rho$ be an $n-$th primitive root of unity, I am interested in a lower bound ...
- 1,020
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151
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A relation concerning the "sum of squares" counting function $r_2(n)$
This is a re-post from MSE as I did not get any response there.
Let $r_2(n)$ denote the number of ways in which a positive integer $n$ can be expressed as the sum of squares of two integers. Here ...
- 2,019
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When can one expect that the $\mu$-invariant of a $\mathbb{Z}_p$-extension of a number field is zero?
What is special about $\mathbb{Z}_p$-extensions which are motivic to ensure that their $\mu$ invariant is zero? Is there a simple conceptual reason.
Here are some examples.
Let $F$ be a totally real ...
user130124
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Approximation of an irrational point from a given direction
Taking norms to be maximal norm, then the simultaneous version of the Dirichlet's approximation theorem states that given real numbers ${\displaystyle \alpha _{1},\ldots ,\alpha _{d}} $,there are ...
- 233
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Remainder term in an integral linked to the Riemann zeta function
Sorry if this is not research level, but the following problem occurs in my own research: it is trivial to show that for $k\ge2$ integral we have
$\zeta(k)=(1/(k-1)!)\int_0^\infty t^{k-1}/(e^t-1)\,dt$ ...
- 11.8k
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What is the sum of the binomial coefficients ${n\choose p}$ over prime numbers?
What is known about the asymptotics, lower and upper bound of the sum of the binomial coefficients
$$
S_n = {n\choose 2} + {n\choose 3} + {n\choose 5} + \cdots + {n\choose p}
$$
where the sum runs ...
- 4,638
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Linear diophantine quasivariety having a unique solution
Consider the equation
$$6x+3y+2z=13$$
for $x$, $y$, $z$ nonnegative integers,
with the constraints
$$x=0\implies y=0,$$
$$x=0\implies z=0.$$
The set of solutions $(x,y,z)$ is a kind of quasivariety
...
- 24.6k
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No rational points on $x^n+a=y^2$ for all $n>4$"?
Is there rational (or better integer) $a$ such that for all $n>4$,$x^n+a=y^2$
has no rational points?
- 24.2k
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Reference Request on logarithm derivative of L-functions
I'm looking for references that show almost all Dirichlet characters $\chi \mod q$ satisfy
$$|\frac{L'}{L}(1+it, \chi)|=o(\log q)$$
where $t\in \mathbb{R}$ is fixed. I have been able to adapt a method ...
- 51
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Isomorphism classes of lattices
Suppose we have a $4 \times 6$ matrix $A$ of rank $4$ whose entries are rational numbers. Define
$$
V = \{x \in \mathbb R^6 \mid A \cdot x = 0\}
$$
and
$$
\Lambda = \{x \in \mathbb Z^6 \mid A \cdot ...
- 1,362
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The Geometry of Jacobi Forms and their Asymptotic Expansions
A Jacobi form of weight $k$ and index $m$ is a meromorphic function $\varphi: \mathbb{H} \times \mathbb{C} \to \mathbb{C}$ satisfying
$$\varphi\bigg(\frac{a \tau + b}{c \tau + d}, \frac{z}{c \tau + d}...
- 1,701
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Reference request: Tate's conjecture for L functions of motives
What's a good reference for the most general form of Tate's conjecture for the order of poles of the L function of a motive? Thanks!
- 1,362
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Is every integer $n>1$ the sum of two triangular numbers and two powers of $5$?
Recall that the triangular numbers are those integers
$$T_n=n(n+1)/2\ \ \ (n=0,1,2,\ldots).$$
In 1796 Gauss proved that each $n\in\mathbb N=\{0,1,2,\ldots\}$ is the sum of three triangular numbers, ...
- 14.5k
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Is there a relationship between the zeta function of a Laplacian and the Selberg Zeta Function?
Let us define the zeta function of an elliptic differential operator $H$ with eigenvalues $\lambda_n$ like so:
\begin{aligned}
\zeta_H(s)
:=
tr( H^{-s} )
\\
:=
\sum^\infty_{n=1} ...
- 447
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113
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$m$-thick sets with small $n$-fold sumsets in finite cyclic groups
Problem. Is it true that for every positive integers $n,m$ there exists a subset $A_{n,m}$ of a finite cyclic group $G$ having the following two properties:
$(\Sigma_n)$ the $n$-fold sum $A_{n,m}^{+...
- 40.9k
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How good are these probabilistic algorithms for the NP-hard problem gcd of sparse polynomials?
The paper NEW NP-HARD AND NP-COMPLETE POLYNOMIAL AND INTEGER DIVISIBILITY PROBLEMS David A. PLAISTED”
defines sparse polynomial as set $\{(a_i,i)\}$ and $f=\sum a_i x^i$.
On p.5:
Theorem 3.3. The ...
- 24.2k
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432
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Why does Faltings' Siegel lemma imply Siegel lemma?
Recall the Siegel lemma:
Let $A = (a_{ij})$ be an $N \times M$ matrix with rational integer coefficients. Put $a = \max_{i,j} |a_{ij}|$. Then, if $N < M$, the equation $Ax = 0$ has a solution $x\in\...
- 3,739
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Does $n$ and $(n-1)/2$ being strong probable primes imply $n$ prime?
A strong probable prime is a positive integer $m$ either $2$, or such that when expressed as $m=d\;2^s$ with $d$ odd, it holds that $2^d\equiv1\pmod m$, or that $2^{(d\;2^r)}\equiv-1\pmod m$ for some $...
- 340
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Modular functions of the type $\mathfrak f(\cdot)^{k}\mathfrak f(\cdot)^{23nk}$
Let $\eta(\omega)$ be the Dedekind eta function and let $$\mathfrak f(w)=e^{-\pi i/24}\frac{\eta((\omega+1)/2)}{\eta(\omega)}.$$
In his paper On the “gap” in a theorem of Heegner, Stark fills the gap ...
- 2,335
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145
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For Hida theory on $GU(2,2)$ can $p$ be inert in the imaginary quadratic field $K$?
I am familiar with the theory of Hida families of modular forms, so Hida theory on $GL_2$, but I am not familiar with Hida theory on any other algebraic groups. My question concerns Hida families of ...
- 323
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Characters on $PGL(2)$
I am interested in computations with characters on $PGL(2, F)$ and not in $GL(2, F)$, and have some issues concerning definitions.
The notion of conductor is standard for characters $\chi$ of a $p$-...
- 5,314