All Questions
Tagged with reference-request nt.number-theory
1,408 questions
0
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Reference request: on sums of the form $ax^m + by^n = h$
I know that equations of the form
$$\displaystyle ax^d + by^d = h$$
with $a,b,h \in \mathbb{Z}$ have been thoroughly investigated as a special (and interesting) case of the Thue-Mahler equation, for ...
- 26.9k
3
votes
1
answer
151
views
Definability of orderings on a formally real number field
For vector basis $b_1,..,b_n$ on a finite extension $F$ of $\mathbb{Q}$, where $-1$ is not a sum of squares, each linear order on $F$ is determined by an order on the basis. This uses information ...
- 11.1k
9
votes
2
answers
1k
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runs of consecutive non squarefree integers
This question gained no attention at Math SE.
Call a sequence of $k$ consecutive naturals squary if each one of them is divided by a square > 1. The Chinese Remainder theorem trivially guarantees us ...
- 13.4k
6
votes
1
answer
121
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Name or references for minimal $N$ such that $\left(\frac{a}{b}\right)_n = \left(\frac{a}{b'}\right)_n$ whenever $b \equiv b' \bmod (N)$
Let $\left( \dfrac{a}{b} \right)_n$ denote the nth power residue symbol, a generalization of the Legendre symbol. I have recently seen it quoted that there is a minimal ideal $N$ (minimal by ideal ...
- 1,570
7
votes
1
answer
652
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Fermat-quotient of "order" 3: I found $68^{112} \equiv 1 \pmod {113^3}$ - are there bigger examples known?
(I've taken this from MSE, it seems to be more appropriate here)
I'm rereading an older text on fermat-quotients (see wikipedia) from which I have now the
Question for
$$ b^{p-1} \equiv 1 \pmod{ ...
- 5,342
2
votes
0
answers
224
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Abel-Jacobi map isomorphism galois representations
Let $X/\mathbb{Q}$ be an irreducible smooth projective curve with a $\mathbb{Q}$-rational point $p$. Then there is a map $\phi: X \rightarrow \textrm{Pic}^{0}(X)$ with the property that $q \mapsto [q]-...
- 3,555
9
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2
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683
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The Theory of Transfinite Diophantine Equations [closed]
The theory of Diophantine equations is one of the main stream research areas in number theory. There are many known results and unknown conjectures about the existence of non-trivial solutions for ...
user42090
5
votes
1
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605
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Who is attributed with the conjecture that every multiply-perfect number greater than $1$ is even?
I know that Descartes is considered to be the first to ask whether or not odd perfect numbers exist ($n$ such that $\sigma(n)=2n$, where $\sigma(n)$ is the sum of divisors of $n$), and he also ...
- 205
13
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2
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644
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Reference for a conjecture on the first primes congruent to 1 modulo other primes
Given a prime $p$, define $f(p)$ to be the smallest prime congruent to $1$ modulo $p$. For example, $f(7)=29$. It has been conjectured that $f(p)<p^2$ always: by Schinzel in his "Hypothesis H" ...
- 12.8k
2
votes
0
answers
236
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Is there a generalization of Granville-Langevin conjecture for number fields?
According to Wikipedia and other sources the Granville-Langevin conjecture
states:
If $f$ is a square-free binary form of degree $n > 2$, then for every real $\beta > 2$ there is a constant $...
- 25.4k
25
votes
1
answer
2k
views
The origin of Discrete `Liouville's theorem'
It is known that discrete Liouville's theorem for harmonic functions on $\mathbb{Z}^2$ was proved by Heilbronn (On discrete harmonic functions. - Proc. Camb. Philos. Soc. , 1949, 45, 194-206).
If ...
- 12.3k
16
votes
1
answer
4k
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Order of magnitude of $\sum \frac{1}{\log{p}}$
Question: What is the order of magnitude of the following sum?
$$ \sum_{\substack{p<n\\\text{$p$ prime}}} \frac{1}{\log{p}} $$
Additional information: Since
$$ \sum_{\substack{p<n\\\text{...
- 3,004
1
vote
2
answers
191
views
Reference request for Frobenius numbers
The Frobenius number of a set of coprime integers is the largest number that not can be written as the sum of integer multiples of numbers in that set.
I'm looking for a general reference on ...
- 563
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4
answers
2k
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Interactions of number theoretic conjectures and other fields of mathematics
There are many interesting open conjectures in number theory. My question is not about partial results or possible ways to prove them. It is about their interactions with the other fields of ...
2
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2
answers
334
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What should I read if I want to learn about integral structures on classical algebraic groups?
I'm looking to learn about integral structures (or models?) on classical algebraic groups.
To begin with I have been learning about algebraic groups, quadratic forms and lattices. And also looking at ...
- 787
3
votes
1
answer
246
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Numbers with balanced diophantine approximations
This is a follow-up to Question 146635, namely Expected symmetry in the diophantine approximations of an irrational number, which I will refer to for notation and terminology used here without ...
- 10.5k
7
votes
1
answer
288
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Expected symmetry in the diophantine approximations of an irrational number
Given $x \in \mathbb{R}$ we will write $\{x\}$ for the fractional part of $x$ and $\|x\|$ for the distance of $x$ from the nearest integer, in such a way that $\{x\} = x - \lfloor x \rfloor$ and $\|x\|...
- 10.5k
10
votes
2
answers
5k
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Cohen-Lenstra Heuristics reference
I am looking for good references (preferably, books) on Cohen-Lenstra Heuristics (on Real Quadratic fields) which explain in detail the reasons behind its fundamental assumption (higher the ...
- 1,305
1
vote
1
answer
323
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Koch's "Extendible functions"
For more than a year now, I have been looking for a copy of the following CICMA Concordia preprint :
Author : Helmut Koch
Title : Extendible functions
Preprint, CICMA Concordia University Department ...
- 18.7k
4
votes
1
answer
590
views
To which automorphic forms/rep's over a function field can we associate a Galois representation?
As far as I understand it, by the work of Lafforgue (cf. Laumon, "Cohom. of Drinfeld ... II", Thm 12.4.1) there is a Galois representation associated to an irreducible cuspidal automorphic ...
- 43
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0
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234
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On the irrationality measure of generalized Stoneham numbers
Pick non-zero integers $a,b,c$ with $a,b \ge 2$ and let $\xi_{a,b,c}$ be the sum of the series $\sum_{n=1}^\infty a^{-b^n} c^{-n}$ (no restriction is made on the sign of $c$); when $b = c$ and $\gcd(a,...
- 10.5k
8
votes
1
answer
620
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On the irrationality measure of $\sum_{n=1}^\infty a^{-b^n}$
Pick integers $a, b \ge 2$ and let $\xi_{a,b}$ be the sum of the series $\sum_{n=1}^\infty a^{-b^n}$. It is known that $\xi_{2,2}$ is transcendental: I learned a proof of this from notes by M. ...
- 10.5k
9
votes
1
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569
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The Dissertation of F. J. van der Linden
Does anyone have access to the 1984 dissertation of Franciscus Jozef van der Linden under Hendrik Lenstra? It is called Euclidean Rings with two infinite primes. The theory is that this has the ...
- 25.7k
0
votes
0
answers
694
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"Descending cohomology, geometrically" by Mazur:
(Exist texts of that talk or related texts: http://ttv.mit.edu/collections/harris60/videos/13881-problem-session-barry-mazur ?) Article: http://www.math.harvard.edu/~mazur/papers/page37.pdf
- 10.8k
9
votes
4
answers
1k
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The relationship between the dilogarithm and the golden ratio
Among the values for which the dilogarithm and its argument can both be given in closed form are the following four equations:
$Li_2( \frac{3 - \sqrt{5}}{2}) = \frac{\pi^2}{15} - log^2( \frac{1 +\...
- 151
4
votes
1
answer
1k
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Books on the Hardy-Littlewood circle method
Are there any good books providing an introduction to the Hardy-Littlewood method that do not require much of a background in complex analysis?
- 1,890
5
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0
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974
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$\sum_{n=1}^{\infty}\frac{1}{a_n}=\infty$ $\sum_{n=1}^{\infty}\frac{1}{b_n}=\infty$ but $\sum_{n=1}^{\infty}\frac{1}{a_n+b_n}=c, c\in R$ [closed]
The following question is inspired from: Defining the slowest divergent series.
Let $a_n$ and $b_n$ be two strictly increasing sequences of natural numbers,with $\sum_{n=1}^{\infty}\frac{1}{a_n}=\...
- 3,866
1
vote
0
answers
272
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Possible counterexample to the strong three exponentials conjecture
There is something wrong possibly either with me or with Wikipedia.
Wikipedia's article on the strong three exponentials conjecture
defines $L^\ast$ as the set of all complex numbers of the form
$$\...
- 25.4k
0
votes
0
answers
215
views
Invariance of the (Liouville-Roth) irrationality measure under rational Möbius transformations
For a real number $x$, we define the (Liouville-Roth) irrationality measure of $x$, here denoted by $\mu(x)$, as the infimum, with respect to the poset $(\mathbb{R}_0^+ \cup \{\infty\}, \le)$, of the (...
- 10.5k
31
votes
5
answers
8k
views
Fermat's proof for $x^3-y^2=2$
Fermat proved that $x^3-y^2=2$ has only one solution $(x,y)=(3,5)$.
After some search, I only found proofs using factorization over the ring $Z[\sqrt{-2}]$.
My question is:
Is this Fermat's original ...
- 3,866
8
votes
3
answers
882
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Reference request for $(1,2^n-1,2^n)$ example related to abc-conjecture
The $abc$-conjecture states that if $a,b,c$ are positive, relatively prime integers satisfying $a+b=c$, then the product of the primes dividing $abc$ (the radical of $abc$) is $\gg_\varepsilon c^{1-\...
- 12.8k
6
votes
3
answers
938
views
Uniformly distributed sequence in $\mathbb{R}$
We say that a sequence $(x_n)_{n=1}^\infty \subseteq \mathbb{R}$ is "uniformly distributed in $[a,b]$", with $a < b$, if $(x_n)_{n=1}^\infty \cap [a,b] \neq \varnothing$ and
$$\lim_{N \to \infty} \...
- 61
6
votes
1
answer
1k
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Using the decomposition $641 = 5^4 + 2^4$ to factor $F_5$
The question in the title arises from a problem in Stewart's "Galois Theory, Third Edition" (and possibly elsewhere) which has been bugging me for a few days since reading it:
Problem 19.5 (p. 224) ...
- 5,191
2
votes
1
answer
219
views
RefReq: Algorithms for standard operations in Algebraic Number theory
Given an algebraic number field $F$ (I actually don't have an idea how to implement this data already, except for splitting fields of polynomials, but there is something in SAGE) is there free code ...
- 11.2k
1
vote
2
answers
505
views
A conjecture of Montgomery: reference request
In the answer to this question, engelbret mentions "a conjecture of H. L. Montgomery (not the one on pair correlations, another one), which implies both the GRH and the Elliott-Halberstam conjecture."....
- 26k
1
vote
0
answers
86
views
Classification of involutions of the lattice $H\oplus H(k)^{\oplus2}$ for $k=5,6$?
Let $H$ denote the hyperbolic lattice (rank 2 lattice generated by $e,f$ such that $e^2=f^2=e.f-2=0$). Let $k >0$ be an integer. Is it possible to classify involutions $\iota$ of the lattice
$$
L:=...
- 1,663
5
votes
1
answer
819
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Is this known alternating sum for Euler's constant?
This probably is known, but Wolfram Alpha doesn't recognize it
and couldn't find it in Mathworld (there is something close,
but using floor).
We have
$\lim_{s \to 1} (\zeta(s)-1/(s-1)) = \gamma$
...
- 25.4k
4
votes
0
answers
242
views
Evaluation of $E_{\ell,2}$ on supersingular curves over $\mathbb{F}_{p^2}$
As mentioned in an answer to Modularity of $E_2$ on congruence subgroups, there exist modular forms $E_{\ell,2}$ of level $\Gamma_{0}(\ell)$ and weight 2, with $q$-expansion $E_{\ell,2}(q)=E_{2}(q)-\...
- 1,537
0
votes
1
answer
236
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Congruences for generalized Franel numbers
Let us define generalized Franel numbers $f^{(m)}_n$ through recurrence relations:
$f^{(1)}_n=1$ for all $n$, and $$f^{(m)}_n=\sum\limits_{k=0}^n\binom{n}{k}^3f^{(m-1)}_k.$$ In fact $$f^{(m)}_n=\sum\...
- 16.5k
1
vote
0
answers
280
views
Would a proof of both (G)RH and Montgomery's pair correlation conjecture imply SOC?
Would a proof of both (G)RH and Montgomery's pair correlation conjecture imply SOC?
It seems, judging by the abstract of a 2002 paper of Ram Murty and a possibly Romanian co-author published on www....
- 6,984
2
votes
1
answer
240
views
Congruence for the Apery Numbers
Is it true that $$A_n\equiv (-1)^n\;\;(\mathrm{mod}\;3)\;\;?$$
Here $A_n$ is the Apery number:
$$A_n=\sum\limits_{k=0}^n\binom{n}{k}^2\binom{n+k}{k}^2.$$
What is known about congruence properties ...
- 16.5k
1
vote
6
answers
1k
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List of structure theorems for vector valued Siegel modular forms (esp. of genus 2)
What are Siegel modular forms?
We start with defining their common domains $\mathbb{H}_g$ as the set of symmetric $g \times g$ matrices with positive definite imaginary parts.
The symplectic group $...
3
votes
0
answers
680
views
Birch/Swinnerton-Dyer "Notes on Elliptic Curves II"
I would like to know if any of you know if there is a more general treatment to what Birch and Swinnerton-Dyer did in "Notes on Elliptic Curves II" (http://www.ams.org/mathscinet-getitem?mr=179168).
...
- 521
4
votes
0
answers
846
views
Reference request for a basic result on relative differents & discriminants
I am looking for a better reference for the results in this extremely short and elementary paper:
Tôyama, Hiraku,
`A note on the different of the composed field',
Kōdai Math. Sem. Rep. 7 (1955), 43–44....
- 1,499
11
votes
1
answer
1k
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The Bombieri Vinogradov Theorem restricted to moduli divisible by $k$
The Bombieri-Vinogradov Theorem states that given $A>0$, there exists $B>0$ such that for $Q=\sqrt{x}\left(\log x\right)^{-B},$ we have $$\sum_{q\leq Q}\max_{y\leq x}\max_{\begin{array}{c}
a\...
- 11.4k
2
votes
2
answers
491
views
Summation of certain series
Suppose $f(n)$ is a periodic function with period $q$. Now from this paper we get that if $\displaystyle\sum_{n=1}^{q}f(n)=0$ then $\displaystyle\sum_{n=1}^{\infty}\frac{f(n)}{n}=-\frac{1}{q}\...
- 1,692
12
votes
1
answer
3k
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Number theory underlying Euler's theory of music
I've recently been studying Euler's theories on music, and I came across Euler's concept of gradus suavitatis or 'degree of pleasure' of a rational number representing the ratio of two tones. (I found ...
- 3,359
8
votes
0
answers
595
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A property of supersingular $j$-invariants (reference request)
Edit 2: For those who understandably don't want to read such a long post, I think Voloch's suggestion reduces the problem to asking whether $j$-invariants of supersingular curves are 3rd powers in $\...
- 1,537
2
votes
0
answers
398
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Counting factors: is this approach in the literature on multiperfect numbers?
Does the following approach (or something near it) exist in the number theory
literature?
I will provide some motivation for $\omega(p^n - 1)$ as $n \rightarrow \infty$
and for this question. First, ...
- 2,158
9
votes
3
answers
980
views
$\omega(p^n - 1)$ as $n \rightarrow \infty$
Although I am also interested in the number of distinct prime factors (not counting
multiplicity), today I use $\omega(m)$ to denote the number of (positive) prime
factors (with multiplicity) of the ...
- 2,158