Prime numbers, Diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

learn more… | top users | synonyms (1)

0
votes
0answers
40 views

Counting number of points in a lattice with bounded length

I am interested in counting number of lattices using the following theorem. The following is Theorem IV (page 412) in Chapter VIII of "An introduction to the geometry of numbers (second printing, ...
10
votes
0answers
127 views

Enriques surfaces over $\mathbb Z$

Does there exist a smooth proper morphism $E \to \operatorname{Spec} \mathbb Z$ whose fibers are Enriques surfaces? By a theorem of, independently, Fontaine and Abrashkin, combined with the ...
1
vote
0answers
61 views

Square-free sieve over number fields

I am trying to work on extending various methods to study square-free values of polynomials (or more generally, $k$-free values) over general rings of integers, and a literature review has yielded ...
-2
votes
0answers
150 views

Proof theory and the generalized Riemann hypothesis [on hold]

Is there a disproof of the following? CONJECTURE: Let $\chi$ be a Dirichlet character modulo $q$. Let $\varepsilon$ be a positive number with $0 < \varepsilon < \frac{1}{2}$. Let $T$ be a ...
-4
votes
2answers
173 views

Brocard's problem [on hold]

According to Brocard's problem $$x^{2}=n!+1 \Rightarrow (x+1)(x-1)=n!$$, where $(x+1)/2$ , $(x-1)/2$ are consecutive integers, assume one of them is z, another is (z-1). z and (z-1) have all primes ...
5
votes
0answers
103 views

Non-embeddable varieties

Suppose that $k$ is a perfect field of characteristic $p>0$, $\mathcal{V}$ is a complete discrete valuation ring with residue field $k$ and quotient field $K$, of characteristic $0$. Then when ...
1
vote
0answers
52 views

On conductors, levels and traces on quaternion algebras

I am currently working on conductor and level issues in the division central simple algebra case, say $D$ over $F$. I would like to verify some well-known relations between the conductor or the level ...
2
votes
0answers
130 views

Is Frobenius on $R^\circ/p$ surjective for general perfectoid rings $R$?

In [1], Propisition 6.1.9(2), it said that if $R$ is a perfectoid ring such that $pR^\circ$ is closed in $R^\circ$ (this includes the case if $R$ is of character $p$, or if $p$ is invertible in $R$, ...
-1
votes
0answers
59 views

Semiprime number theorem with small prime factor

Hardy & Wright, Theorem 437 gives a nice asymptotic for $k$-almost primes less than $x$. Can we say anything if we restrict one of the prime factors of our almost prime to having a small prime ...
5
votes
1answer
258 views

Etale cohomology approach on $\tau(n)$

Ramanujan's $\tau$ conjecture states that $$\tau(n)=O_\epsilon(n^{\frac{11}2+\epsilon}),$$ which is a consequence of Deligne's proof of Weil conjectures. Answers in ...
1
vote
1answer
69 views

Source for equations involving congruences of Fibonacci and Lucas numbers

In a paper of Cohn (see here), he uses some formulae involving congruences of Lucas- and Fibonacci-numbers (equations 11,12,13 in the preliminaries section). Does anyone know a source for these (and ...
1
vote
1answer
124 views

Does there exist an integer that is both solitary and almost perfect?

This question is an offshoot from the following MSE post. I hope that it is appropriate for this site. Let $\sigma(x)$ be the sum of the divisors of $x$. An integer $a$ is said to be solitary if ...
8
votes
0answers
93 views

Nekrasov-Okounkov hook length formula

I am now reading the paper An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths by Guo-Niu Han. The author rediscovered what he calls ...
3
votes
2answers
246 views

Adeles and twisted adeles

Let $\mu_n$ denote the group of $n$-th roots of unity in ${\mathbb{C}}$, i.e., $\mu_n=\ker[{\mathbb{C}}^*\overset{n}{\longrightarrow}{\mathbb{C}}^*]$. We set $$ \mu=\varinjlim_n \mu_n\subset ...
1
vote
1answer
268 views

Numbers $n$ such that the sum of the divisors of $n$ is a nontrivial power

Let $\sigma (n)$ be the sum-of-divisors function. For example, $\sigma(7)=1+7=2^3$. I know some results about triplets of positive integers $(n,a,b)$ where $a,b\ge 2$ such that $\sigma (n)=a^b$, but ...
-1
votes
0answers
42 views

The product of the power and the natural number in the short interval [on hold]

It is obvious that if $a,b,x\in\mathbb{N}$ and $a^n\leq 2x+1$ then there exists $b\in\mathbb{N}$ such that $a^nb\in\left[x^2,(x+1)^2\right]$. For example for $n=3$, $a=2$ and $x=4$ we have $b=2$ and ...
0
votes
0answers
99 views

Solving the transcendental equation $Li_{3}(e^{-kx}) + x\, Li_{2}(e^{-kx}) = k\, x^3$

I need to solve the following equation: $Li_{3}(e^{-kx}) + x\, Li_{2}(e^{-kx}) = k\, x^3$ for $x\in\mathbb{R}^{\ast}$ and where $k\in\mathbb{R}^{+}$. Here $Li_{3}$ and $Li_{2}$ are the third and ...
5
votes
0answers
114 views

Factorization of antiderivative of minimal polynomials

In any totally real number field, is there an element whose minimal polynomial has the property that its antiderivative factors completely over the rationals? (I’ll let you choose whichever constant ...
3
votes
1answer
193 views

Addition law for elliptic curves of the form $x^2y^2+a(x+y)+b=0$

Did anybody consider addition law for elliptic curves of the form $$x^2y^2+a(x+y)+b=0\,?$$ Does this form have any specific name?
8
votes
2answers
226 views

Root criterion for polynomial over number fields

It's well known that if $\alpha $ is a rational root to an integer coefficient polynomial, then its denominator divides the leading coefficient and its numerator divides the constant term. I'm asking ...
10
votes
2answers
556 views

A “better” rational approximation of pi?

$355/113$ is a good fractional approximation of $\pi$, because we use six digits to produce seven correct digits of $\pi$. $$\frac{355}{113} = 3.1415929\ldots$$ Let $R$ be the ratio of the number of ...
3
votes
0answers
162 views

Density of primes of degree one in Bauer's Theorem (Application of Chebotarev Density)

Let $L$ be a Galois extension of $\mathbb{Q}$ and $M$ a finite extension of $\mathbb{Q}$, both of degrees $> 1$. A Theorem of Bauer tells that $Spl_1(M)\subset Spl(L)$ up to a finite number of ...
13
votes
4answers
1k views

Is there a natural bijection from $\mathbb{N}$ to $\mathbb{Q}$?

In a conversation where it came up that the Pythagoreans probably found an enumeration of the rational numbers I erroneously remarked that Georg Cantor found a natural bijection from $\mathbb{N}$ to ...
2
votes
0answers
71 views

On uniform or simple convergence of Poisson Summation formula

Under good conditions on an even function $f(x)$ we have the Poisson Summation formula ($x>0$): $$f(0) + 2 \sum\limits_{n =1}^{\infty} f(nx)= \frac{1}{x} \left( \hat{f}(0) + 2 \sum\limits_{n ...
4
votes
1answer
236 views

$K_0$ of integral group ring of cyclic group $\mathbb{Z}/p\mathbb{Z}$

Is there a table for the computation of $K_0(\mathbb{Z}[\mathbb{Z}/p\mathbb{Z}])$? These groups are also known as ideal class group in number theory.In topology,they are the home of some important ...
-2
votes
0answers
40 views

related to division algorithem [closed]

Youve seen that the division algorithm says that for m; n, there exists unique q; r such that m = nq + r. Suppose that m = nq + r. What does the division algorithm yield when -􀀀m is divided by b? ...
5
votes
0answers
217 views

Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$

About a month ago, I got the following : For every positive rational number $r$, there exists a set of four positive integers $(a,b,c,d)$ such that ...
3
votes
1answer
77 views

What is the Complexity Class of the “Function Variant” of the Integer Factorization Problem?

I've been reading up a lot Prime Factorization and it's complexity, including a fair number of questions on this very site. However, I still feel there is a question still left unanswered. So, ...
0
votes
0answers
91 views

seminar about the strong multiplicity one for the Selberg class

Very recently, a seminar took place in Seoul with Haseo Ki as an invited speaker to talk about the strong multiplicity one theorem for the whole Selberg class that he did manage to prove. I would like ...
-3
votes
0answers
128 views

Isomorphisms between different models of elliptic curves

As is known, an elliptic curve $E/\mathbb{Q}$ can be represented by a Weierstrass equation of the form $$\displaystyle y^2 = x^3 + Ax + B, A,B \in \mathbb{Z}$$ which is unique provided that if for any ...
0
votes
0answers
107 views

Q re Kaprekar's fixed mapping points

Jens Kruse Andersen in his comment in OEIS's A099009 noticed 3 families of numbers among Kaprekar's fixed mapping points (otherwise known as kernels of the Kaprekar's routine): "Let $d(n)$ denote ...
2
votes
0answers
95 views

Infinite sums with Mobius Inversion : can we have uniform convergence of inversion formula?

My question is on Mobius inversion formula convergence/properties when used with infinite sums of function. Lets consider (on $\mathbb{R}^{+}$): $$S(x)= \sum\limits_{n=1}^{\infty} f(nx)$$ We call ...
4
votes
0answers
123 views

Twists of projective automorphisms

Let $X$ be a projective variety over a perfect field $k$. Recall that a twist of $X$ is a variety $Y$ over $k$ such that $$X_{\bar k} \cong Y_{\bar k}.$$ The twists of $X$ are classified by the Galois ...
0
votes
1answer
155 views

Lower Bound on “exponential” sum

Let $\tau(n)$ be the divisor function. Let $a$ be either a constant, or a function of $X$ that is slowly varying with $X,$ say $X/\log(X)<a(X)<X \log(X),$ for example. I want to lower bound sums ...
0
votes
0answers
87 views

Ward's formula for elliptic divisibility sequences

M. Ward in his Memoir on elliptic divisibility sequences proved that the sequence $\{a_n\}$ defined by recurrence $$a_{n+2}a_{n-2}=a_2^2a_{n+1}a_{n-1}-a_1a_3a_n^2$$ and initial conditions ...
4
votes
1answer
412 views

Algebraic Number Theory in Financial Mathematics

I am currently doing my masters studies in financial mathematics. However, I have had a good background in number theory and I don't feel like leaving it just like that. I am thus inquiring on any ...
2
votes
1answer
315 views

Can you find squares in this class?

For a problem I am working over, I would like to prove that numbers of the following type are not squares $p(l^4+6l^2m^2-3m^4)$ where $p,l,m$ are integers an $p$ prime. I have already found various ...
7
votes
1answer
362 views

Which kind of subsets of primes one needs to generate a positive ratio of the natural numbers?

Not knowing elementary number theory well, I ask this one, which is not very clear to answer, rather I am looking for some results around this question or known theorems. The problem is the following: ...
5
votes
2answers
320 views

Obstruction and rational points on curves

Is etale-Brauer the only obstruction to the existence of rational points on projective plane curves over number fields?
1
vote
0answers
41 views

Number of solutions of arithmetic function's equation [migrated]

Say, an equation is given below \begin{equation} 2\pi(x) - \pi(2x)=\omega(x) \end{equation} where $x$ is a positive integer, $\pi(x)$ is the prime-counting function, and $\omega(x)$ is the number of ...
22
votes
0answers
665 views

Grothendieck's Period Conjecture and the missing p-adic Hodge Theories

Singular cohomology and algebraic de Rham cohomology are both functors from the category of smooth projective algebraic varieties over $\mathbb Q$ to $\mathbb Q$-vectors spaces. They come with the ...
4
votes
1answer
181 views

Quotient of principal congruence subgroups

This is a direct follow-up to this question. What is the quotient $\Gamma(2)/\Gamma(2^n)?$ (the principal congruence subgroups are in $SL(2, \mathbb{Z}).$ It is a 2-group, but what else?
5
votes
4answers
659 views

Consequences of the Inverse Galois Problem

Are there any papers written about the consequences of the Inverse Galois Problem in case it is proved to be true or false? We know a lot of things that would be true if the Riemann Hypothesis holds. ...
5
votes
0answers
102 views

How to assess the influence of a specific term in this telescoping series for $\zeta(s)$?

I like to expand on this (unanswered) MSE question. Take the following, nicely symmetrical, telescoping series for $\zeta(s)$: $$\displaystyle \zeta(s) = \frac{1}{2\,(s-1)} \left(1+\sum ...
1
vote
0answers
59 views

Characterization of two sets

I posted this question on math.SE 10 days ago and had no answer, comment or vote. If an answer is not available, I could really use a reference point as well. For the sake of completeness, I am ...
0
votes
0answers
52 views

Is there any criterion when an interger solution of the equation $ax^2-by^2=c$ $(a,b,c\in\mathbb{N})$ exists? [migrated]

As title. In particular I'd like to know if it suffices that it's solvable in $\mathbb Z_p $ for $p$ being any prime number.
0
votes
0answers
130 views

For which pairs of distinct positive primes $p$ and $q$, the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\sqrt{pq}]$ is a UFD?

For which pairs of distinct positive primes $p$ and $q$, the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\sqrt{pq}]$ is a UFD? I've proved that neither $p$ nor $q$ can be congruent to $1$ modulo ...
2
votes
0answers
99 views

algorithm to find a new point of small height in a number field extension

By the height of an algebraic number $\alpha$, I mean the absolute, logarithmic (additive) Weil height $h(\alpha)$; e.g. $h(2^{1/n}) = (\log 2)/n.$ If $K$ is a number field, let $\delta(K)$ denote ...
2
votes
1answer
102 views

Enumerating positive fractions (reference missing)

I remember that the recursion $r(0)=0, \ \ r(n+1)=\frac{1}{2 [r(n)]+1-r(n)}$ produces a sequence of rational values $ 0 \mapsto 1 \mapsto 1/2 \mapsto 2 \mapsto 1/3 \mapsto ... $ which exausts the ...
5
votes
1answer
188 views

Complete L-function and FE of Rankin-Selberg on GL(2)?

Let $f$ be a Maass cusp form of $\Gamma_0(N)$ on the upper half plane with character $\chi$ mod $N$ and eigenvalue $1/4+\mu^2$. What is the complete $L$-function of the Rankin-Selberg product ...