**2**

votes

**1**answer

278 views

### Constant group scheme and torsors

Let $X$ be a scheme and $G$ a (commutative) constant group scheme. Consider a $G$-torsor $Y$ for $X$, by which I mean that there is a canonical isomorphism:
$$g_Y \colon Y \times_X Y \cong Y ...

**12**

votes

**0**answers

233 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 ...

**11**

votes

**3**answers

528 views

### Counting points on lattices

I expect that the following is a standard problem from analytic number theory, but I don't know where exactly to look for an answer.
Let f: ℤr→ H be a surjective homomorphism into a ...

**2**

votes

**0**answers

67 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, ...

**-4**

votes

**2**answers

188 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 ...

**13**

votes

**2**answers

747 views

### Simple proofs for the existence of elliptic curves having a given number of points

Yesterday, after he gave a nice talk, Dick Gross and I were chatting and he brought up the following annoying problem: suppose that for $p$ a prime that $H_p$ is the "Hasse interval" $[p+1- 2 ...

**1**

vote

**0**answers

66 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 ...

**10**

votes

**2**answers

564 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 ...

**-2**

votes

**0**answers

164 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 ...

**29**

votes

**2**answers

879 views

### Does an existence of large cardinals have implications in number theory or combinatorics?

Does an existence of large cardinals have implications in more down-to-earth fields like number theory, finite combinatorics, graph theory, Ramsey theory or computability theory? Are there any ...

**2**

votes

**0**answers

133 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 $R$, ...

**5**

votes

**0**answers

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

**1**answer

280 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**

vote

**0**answers

54 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 ...

**20**

votes

**15**answers

5k views

### Applications of finite continued fractions

I know some applications of finite continued fractions. Probably you know more. Can you add anything? (For for Applications of periodic continued fractions I have made a special topic.)
1) (Trivial) ...

**5**

votes

**1**answer

261 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**

votes

**0**answers

60 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 ...

**1**

vote

**1**answer

71 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

**1**answer

129 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 ...

**5**

votes

**4**answers

664 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. ...

**3**

votes

**1**answer

194 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?

**4**

votes

**1**answer

206 views

### Constructing a family of convergents from continued fractions formed by a set of prime partial quotients

For a given real number $x$, the continued fraction representation $x = [a_0; a_1, a_2, \cdots]$ where $(a_n)_{n \geq 0}$is defined by setting $x = \alpha_0$, then $a_i = \lfloor \alpha_i \rfloor$, ...

**0**

votes

**1**answer

156 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 ...

**13**

votes

**4**answers

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 ...

**9**

votes

**0**answers

300 views

### Can an abelian variety/Q have no non-trivial points over Q_sol?

Let $A/\mathbb{Q}$ be an abelian variety. Must there be a finite solvable
extension $K/\mathbb{Q}$ such that $A(K)$ is nontrivial?
This follows from the conjecture that the maximal ...

**8**

votes

**0**answers

95 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

**2**answers

250 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**

votes

**0**answers

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

**0**answers

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

**0**answers

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 ...

**4**

votes

**2**answers

595 views

### How to calculate the sum of remainders of N?

I'm trying to sum the remainders when dividing N by numbers from 1 up to N
$$\sum_{i = 1}^{N} N \bmod i$$
It's easy to write a program to evaluate the sum if N is small in O(N) but what if N is large ...

**10**

votes

**8**answers

4k views

### Is there an algorithm to solve quadratic Diophantine equations?

I was asked two questions related to Diophantine equations.
Can one find all integer triplets $(x,y,z)$ satisfying $x^2 + x = y^2 + y + z^2 + z$? I mean some kind of parametrization which gives all ...

**4**

votes

**0**answers

124 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 ...

**8**

votes

**2**answers

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 ...

**0**

votes

**0**answers

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 ...

**6**

votes

**1**answer

319 views

### A conjecture related to the Cohen-Oesterlé dimension formula of spaces of modular forms for half-integer weights

Denote $\operatorname{dim}(M_k(\Gamma_0(N)))$ by $m(k,N)$ and
$\operatorname{dim}(S_k(\Gamma_0(N)))$ by $s(k,N)$.
Let $N$ any positive multiple of $4$ and $j \ge 1$.
$$
a(N) := \frac1j \left(m ...

**3**

votes

**0**answers

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 ...

**5**

votes

**1**answer

194 views

### Optimal lower bounds for the sum of digits in base $b$

Let $b \geq 2$ be an integer and let $s_b(n)$ be the sum of the digits of the base-$b$ representation of the nonnegative integer $n$
(e.g., $s_{10}(726)=7+2+6$). From the weak law of large numbers, it ...

**2**

votes

**0**answers

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 ...

**3**

votes

**1**answer

714 views

### Efficient algorithms to determine the roots of: $p(x) = r^x $ in the finite field $GF(q)$, where $r$ a primitive root of the field

I need to make sure that no efficient (i.e., polynomial time) algorithm exists for the following problem:
Exponentiating Polynomial Root Problem (EPRP)
Let $p(x)$ be a polynomial with $\deg(p) \geq ...

**4**

votes

**1**answer

239 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

**0**answers

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?
...

**3**

votes

**1**answer

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, ...

**5**

votes

**0**answers

221 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

**1**answer

821 views

### Exact formula for the number of integers in an interval which are the sum of two squares.

Denote by $\lambda(n)$, the number of numbers between $0$ and $n$ which are the sum of two squares. Landau, and Ramanujan have proven independently, that $$\lambda(n) \sim \frac{n}{\sqrt{\ln(n)}}$$
...

**7**

votes

**1**answer

476 views

### Which L-functions are not “Langlands-Shahidi L-functions”?

The Langlands-Shahidi method, among other things, obtains certain L-functions from the constant term of Eisenstein series attached to so-called $(G,M)$ pairs, where $G$ is a reductive group, $M$ a ...

**0**

votes

**0**answers

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 ...

**2**

votes

**0**answers

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 ...

**-3**

votes

**0**answers

129 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

**0**answers

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 ...