All Questions
Tagged with reference-request nt.number-theory
1,408 questions
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On a probabilistic integer factorization algorithm given bounds for one prime factor
We got a probabilistic integer factorization algorithm and experimental evidence with large
integers given bounds for one factor.
Let $D \ge 2$ be real number and let $p,q$ be primes and $N=pq$.
...
- 25.4k
1
vote
0
answers
267
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Are there connections between Calabi-Yau manifolds and number theory?
I am interested in understanding whether there are any significant connections between Calabi-Yau manifolds and number theory. Calabi-Yau manifolds are central objects in algebraic geometry and string ...
8
votes
1
answer
677
views
Infinite series and sum of two squares
Consider the following infinite sequence $a(n)$ generated by
$$\sum_{n\geq0} a(n)q^n
=\frac{\sum_{k\geq0}F(2k+1)q^{\binom{k+1}2}}{\sum_{k\geq0} q^{\binom{k+1}2}}$$
where the $F(2k+1)$ are the odd ...
- 43.2k
2
votes
1
answer
215
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Representing positive integers $n$ by binary forms $n=ax^2+by^2$, $a\geq 0$, $b\geq 0$
In there a finite set of $(a_k,b_k)\in\mathbb{Q}_{\geq 0}\times \mathbb{Q}_{\geq 0}$ so that any $n\in \mathbb{Z}_{>0}$
can be written as $n=a_k x^2+b_k y^2$ for some $k$ ?
This is related to ...
- 14k
-3
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1
answer
195
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Bounding a number-theoretic integral
Find a good upper bound on $$\int_1^T\frac{\zeta'(s)}{\zeta(s)\zeta(1-s)}X^sdt,$$ where $s=c+it$ for a constant $c>1$ and $X>0$ is a parameter. If needed, we can assume RH.
My attempt here is ...
- 107
23
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4
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2k
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Identity for an infinite product
Here is an experimental "result" exhibiting the difference of two (formal) infinite products that "almost factorizes".
QUESTION. Is this true?
$$\prod_{n\geq1}(1+x^{2n-1})^{24} - \...
- 43.2k
0
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0
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140
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State of the art on attempts to solve the elliptic curve discrete logarithm problem through transfering the problem to a weaker curve
Let an elliptic curve $E$, and 2 points on such curve $P$ and $O$ the methods I’m talking about consist in creating a weaker elliptic curve $F$ and mapping $P$ and $O$ to $F$ while successfully ...
- 87
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2
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3k
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Is it known? A sum over lattice parallelograms of area one is equal to $\pi$
I recently discovered a formula, my proof is really a high school proof in three lines.
$$4\sum_{x, \, y \, \in \, \mathbb Z_{\geq 0}^2, \, \det(x \ \ y) = 1} \frac{1}{\lVert x\rVert^2\cdot\lVert y\...
- 5,063
1
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1
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241
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The equation $ax^2 +by^2 =1 \mod P$ in cyclotomic field
Let $L$ be a cyclotomic field, and $P$ a prime ideal of $\mathcal{O}_L$.
is there any symbol for the equation $ax^2 + by^2 =1 \mod P$ and if so, is it computable in polynomial time?
if $a$ is ...
- 133
71
votes
8
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12k
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Possible new series for $\pi$
In a recent (unfortunately over-hyped) preprint by Saha and Sinha, Field theory expansions of string theory amplitudes (arXiv:2401.05733), they present the following series for $\pi$:
$$\pi = 4 + \...
- 82.7k
3
votes
1
answer
123
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Reference request: ray class group as quotient of finite ideles
Let $K$ be a number field, and write $\mathbb{A}_{K,f}^\times$ for the group of finite ideles of $K$. That is
$$
\mathbb{A}_{K,f}^\times = \{(u_v)_v \in \prod_{v \nmid \infty} K_v^\times : v(u_v) = 0 \...
- 411
6
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1
answer
407
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Good reduction for the universal elliptic curve
Let $X$ be a modular curve, i.e. a quotient of the upper half plane $\mathbb{H}$ by a congruence subgroup $\Gamma$. When $\Gamma=\Gamma_0(N)$, it is known that $X$ has a smooth model denoted $\mathcal{...
- 2,907
1
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0
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150
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What are alternative mathematical definitions of observers beyond Bennett and Hoffman's framework?
Motivation:
This question is inspired by a talk from Avi Wigderson given on Randomness, where the idea that the randomness is in the eye of the observer is suggested.
In the study of information ...
- 3,074
1
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0
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45
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Proper Pisot n-tuples
Recall that x is a Pisot number if it is real and x>1, while all of its conjugates have magnitude less than 1. Then $\{(x)^k\}$ (where $\{\cdot\}$ is the fractional part of x) approaches 0 ...
- 680
2
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1
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431
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Shadows of partitions of lcm
$\DeclareMathOperator\lcm{lcm}$Fix an integer $n\geq1$. Denote the least common multiple $L_n=\lcm(1,2,\dots,n)$.
QUESTION. Is the following true? For each integer partition $\lambda=(\lambda_1,\...
- 43.2k
10
votes
0
answers
288
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Coefficients of polynomials vs trigonometric product
Let's consider the family of sequences of coefficients in the expansion
$$\prod_{i=0}^{n-1}(1+x^{3^i}+x^{3^{i+1}})=\sum_{k\geq0}a_n(k)\, x^k.$$
Remark. Evidently, the RHS is a finite sum.
Here is a ...
- 43.2k
2
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2
answers
363
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Size of $\zeta'(s)$ at its zeros
How large can the derivative of the Riemann zeta function be at its zeros?
More specifically, let $\rho$ be a zero of the zeta function with $\Im(\rho)\in (0,T]$. What can we say about $|\zeta'(\rho)|...
user536681
1
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1
answer
222
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Sum over three squares
Let $x$ be a sufficiently large number. Is there an explicit or asymptotic formula for the following sum
$$\sum_{\substack{n\leq x\\ n=a^2+b^2+c^2}} 1.$$ Any reference would be helpful.
- 1,257
3
votes
3
answers
385
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On subfields of the cyclotomic field $\mathbb{Q}(\zeta_p)$
Let $p$ be an odd prime. Let $\zeta_p=e^{2\pi{\bf i}/p}$ and let $1\le k\le p-1$ be a divisor of $p-1$. Recently, when I learnt algebraic number theory, I met the following problem.
If we let
$$U_k=\{...
- 87
16
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3
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4k
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Is it known that the Collatz-like sequence with 7n+1 diverges to infinity starting with 7?
In this question I was wondering if the $3$ in the Collatz conjecture is arbitrary, and when I wrote that question I tried to change to $7n+1$ starting with the seed number $7$, the sequence appears ...
- 541
2
votes
1
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186
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On local Galois deformation rings
Let $p,\ell$ be two different primes. Let $K$ be a finite field extension of $\mathbb{Q}_{\ell}$ and $ \bar{\rho}:G_{K}\to {\rm GL}_{n}(\mathbb{F}_p) $ be a continuous mod $p$ representation of the ...
- 667
42
votes
4
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4k
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Are these fast convergent series for $\log(2)$, $\log(3)$ and $\log(5)$ already known and proven?
Now that some of the previously MSE formulae that I left here have been applied Dec.2023 to compute high precision record values ($10^{12}$ decimal digits) of trascendental constants $\Gamma(1/3)$ (Eq....
- 2,836
3
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0
answers
149
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Solutions of a quadratic Diophantine equation over algebraic integers
The problem is not very exactly Diophantine in the classical way. I am trying to find some algebraic integer solutions in a number ring to a quadratic equation over the same ring.
Precisely, let $\...
- 287
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0
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156
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Reference Request: Completeness of the space of all Whittaker models(a lemma in JPSS1981)
$\DeclareMathOperator\GL{GL}$There is a lemma in the proofs of local converse theorem stated as
Suppose $F$ is a non-archimedean local field, $\psi$ is a non-trivial addtive character on $F$. $N_n$ ...
- 53
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0
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101
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Identities for Prime Coefficients of Certain Cusp Forms
While working with Fourier expansions of cusp forms of congruence subgroups of the modular group, I observed the following patterns in their prime coefficients.
Let $a(n)$ be the Fourier coefficients ...
5
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0
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262
views
Primes generated by cyclotomic polynomials
Let $p$ be an odd prime, and let $f=\Phi_p$ be the $p$-th cyclotomic polynomial. Denote by $S_p$ the set of primes $q$ such that there exists a sequence of primes $p_1,\dots, p_g$ such that $p_1=f(1)=...
2
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1
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191
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Sums of multiplicative functions over residue classes
It was stated in this Shiu, P. work, page 169, Theorem 2, that $$\sum_{\substack{n\le x\\ n\equiv a\pmod k}}d_r^{\ell}(n)\ll\frac{x}{k}\left(\frac{\phi(k)}{k}\log x\right)^{r^{\ell}-1}.$$
Here, $d_r(n)...
user536681
15
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3
answers
1k
views
Does anyone remember what happened to the experimental search for polynomial identities for $\pi$?
So a while back I was on the internet and had encountered a website containing an experimental search for identities for $\pi$. My memory was that the page belonged to either Jonathan Sondow or ...
- 2,689
10
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0
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351
views
How are the hypergeometric motives of WZ-Pairs connected?
If $\small{(F,G)}$ is a WZ-pair and general asymptotic conditions $\lim_{k\rightarrow\infty}\small{G(n,k)=0}$ and $\lim_{n\rightarrow\infty}\small{F(n,k)=0}$ hold, then we have the certified ...
- 2,836
3
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2
answers
342
views
Reference Request: Beilinson-Bloch conjecture in terms of Beilinson regulator isomorphism
I'm looking for a reference that provides a concise statement of the Beilinson-Bloch conjecture, specifically formulated in terms of an isomorphism under the Beilinson regulator map.
More precisely, I'...
- 2,907
4
votes
1
answer
165
views
Reference Request: on an explicit formula for class-1 Whittaker functions on split reductive groups over p-adic fields
The 1978 preprint by S.Kato 'On an explicit formula for class-1 Whittaker functions on split reductive groups over p-adic fields' is cited by papers involving unramified computations of local ...
- 43
12
votes
1
answer
238
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Number of planes generated by integer vectors
For fixed dimension $d$ and large $R$ consider all non-zero integer vectors in the ball $B(0,R)\subset \mathbb{R} ^d$ of radius $R$ centered at the origin. The number of such vectors grows as $c_d\...
- 109k
5
votes
1
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311
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Is there a statement in Presburger arithmetic about primes this simple heuristic fails for?
I came up with the following conjecture while thinking about ways to formulate some heuristics about primes:
Conjecture: Given a statement $s$ in Presburger arithmetic, using an additional unary ...
- 3,319
8
votes
2
answers
178
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Distribution of traces and max entries of words of fixed length in $\operatorname{SL}_2(\mathbb{N})$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\tr{\mathsf{tr}}$$\SL_2(\mathbb{N})$ is a free monoid on the generators
$$
L=\begin{pmatrix}1&0\\1&1\end{pmatrix},\quad R=\begin{pmatrix}1&1\...
- 609
1
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1
answer
175
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Original text about Riesel numbers?
Riesel numbers appeared in the answer to my question
Solving $2^{x+1} m -1=p^y$ for prime $p$ and natural $x,y$.
The original reference
Riesel, Hans (1956). "Några stora primtal". Elementa. ...
- 14k
2
votes
1
answer
101
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Bound on number of extensions of Q unramified outside a fixed prime
Are there any known asymptotic bounds on the number of degree $d$ extensions of $\mathbb{Q}$ unramified outside a fixed prime $p$?
- 2,907
0
votes
2
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216
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Papers related to a diophantine equations about Magic square of squares for $n=3$
The open problem of magic squares of squares explained here. Consider the following magic square of squares:
$$
\begin{aligned}
&a^2&b^2&&c^2\\\\
&d^2&e^2&&f^2\\\\
&...
3
votes
0
answers
79
views
Reference request for unitary Shimura varieties
Let $K$ be an imaginary quadratic field and let $X$ be the Shimura variety associated to the unitary group $U(m,n)$ over $K$ (after a suitable choice of PEL datum).
Is there a reference that explains ...
- 1,949
4
votes
0
answers
168
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Explicit bounds on gaps between zeros of $\zeta^\prime(s)$
In $\S$9.1 of "Theory of the Riemann Zeta Function", Titchmarsh uses Borel-Carathéodory and Hadamard Three Circles to show that every circle of radius 6 and center $3+iT$ contains a zeros of ...
- 11.1k
1
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1
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142
views
Algorithm for computing isogeny class of elliptic curve
Is there an algorithm for computing the entire isogeny class of a given elliptic curve $E/\mathbb{Q}$?
References/ideas are welcome. Thanks!
- 2,907
16
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1
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357
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Galois cohomology for non-Galois extensions
If $L/K$ is a Galois extension with group $G$ then we can consider $H^*(G;L^\times)$. This is useful in algebraic number theory, and there are many results about it.
Now let $L/K$ be a finite ...
- 56.9k
0
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108
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looking for reference for two elliptic curves with equal formal group
I am looking for a reference.
In this post, @Chris Wurthrich made the following comment:
If the formal group laws (probably upon particular choice of coordinates) of two elliptic curves over any ring ...
- 195
2
votes
0
answers
179
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A Brun-Titchmarsh type result for divisor sums; asymptotic/improved bound
In Shiu's work ('A Brun-Titchmarsh theorem for multiplicative functions') he proved that if $r\le x$ is a natural number, we have $$\sum_{r<n\le x}d(n)d(n-r)\ll x\log^2x\sum_{d|r}\frac{1}{d}.$$
I ...
user344548
8
votes
0
answers
208
views
Elliptic curves of rank 1 over number fields
I am interested what is known about the following statement:
For every number field $K$, there exists an elliptic curve $E$ defined over $K$ with algebraic rank equal to $1$.
Is this statement known ...
- 502
3
votes
0
answers
148
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Casimir eigenvalues of p-adic automorphic representations
In the context of p-adic local Langlands correspondence:
Is it possible to define Casimir eigenvalues for p-adic automorphic representations? If a local representation arises from a global Galois ...
- 2,907
0
votes
1
answer
223
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Reference for Faltings' proof on finiteness of semisimple $d$-dimensional $p$-adic Galois representations
I'm looking for a reference to Faltings' proof concerning the finiteness of $d$-dimensional semisimple $p$-adic Galois representations. Specifically, the result states that there are only finitely ...
- 2,907
1
vote
0
answers
25
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Characterising rank-$2$ lattices $\Lambda$ and conjugate-linear translate $g \sigma(\Lambda)$, given elementary divisors
Let $E/F$ be a quadratic unramified extension of local fields with $\operatorname{char} F = 0$. Let $\Lambda \subseteq E^2$ be an $O_E$-lattice of rank $2$. Let $g \sigma \in \operatorname{GL}_2(E)$ ...
- 111
6
votes
2
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755
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Prove positivity of a binomial sum
Some problems appear easy on the face of it, but perhaps they are not. Here is an instance of a certain calculation which is slightly reformulated from its original encounter in a current work. I have ...
- 43.2k
1
vote
2
answers
296
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Possible refinements of the large sieve inequality
Let $a_n$, $1\leq n\leq N$, be complex numbers, and set $S(\alpha)=\sum\limits_{n=1}^{N}a_ne(n\alpha)$, where $e(\alpha)=\exp(2i\pi\alpha)$. Then, Selberg's large sieve inequality says that
$$\sum\...
- 309
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5
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751
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The diophantine equation $ \sum_{n=1}^{N} \frac{1}{x_{n}} = \prod_{k=1}^{N} \left(1-\frac{1}{x_{k}} \right) $
Background
I wonder if there are any rational numbers such that their Egyptian fraction (sum) representations are equal to their Egyptian product analogue. In other words, I am curious1 about ...
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