**0**

votes

**0**answers

76 views

### Rationality of intersection of algebraic groups

Suppose that $G$ (defined over $\mathbb{Q}$) and $H$ (defined over $\mathbb{R}$) are two algebraic subgroups of a larger algebraic group defined over $\mathbb{Q}$. Assume that $G(\mathbb{R})$ and ...

**0**

votes

**1**answer

153 views

### Normal basis with cyclotomic units

Let p be an odd prime integer and let $\zeta$ be a primitive p-th root of unity.
Let $\alpha$ be a non-trivial cyclotomic unit of $\mathbb Q(\zeta)$, i.e. an element of the form ...

**0**

votes

**0**answers

46 views

### On OPNs and SOPNs

(I hope that this question is appropriate for this site. If it is not, please feel free to point it out and I will then cross-post to MSE.)
OPNs are odd perfect numbers. SOPNs are spoof odd perfect ...

**-1**

votes

**1**answer

1k views

### A stronger version of Fermat's last theorem [duplicate]

Motivated by Fermat's last theorem, one may wonder the following conjecture is true or not.
The equation $x_1^m+\cdots+x_n^m=1$ has nonzero rational solutions iff $n\geq m$.
Here a nonzero rational ...

**6**

votes

**0**answers

231 views

### Can integers be distorted to make primes more regular?

Given a set $P$ of real numbers $\ge 1$, define the gap among different products in $P$ as
$$g(P) = \inf \big\{\prod_{i=1}^n p_i^{a_i} - \prod_{i=1}^n p_i^{b_i} \mid p_i\in P;\,\, p_i\ne p_j \,\text{ ...

**3**

votes

**1**answer

166 views

### When two Dedekind sums are equal

The (classical) Dedekind sum $s(h,k)$ is defined as
$$s(h,k)=\sum_{r=1}^{k-1}\frac{r}{k}\bigg(\frac{hr}{k}-\Big[\frac{hr}{k}\Big]-\frac{1}{2}\bigg)$$
for $\gcd(h,k)=1$.
A natural question is, when ...

**2**

votes

**2**answers

207 views

### $t$-analogue of the symmetric power of an additive character over $\Bbb{F}_q^*$

Let $G$ be a finite group and let $f: G \longrightarrow \Bbb{C}$ be any complex-valued function. For integers $k, n \geq 0$, an indeterminant $t$, and $x \in G$ let $f_k(x) := f \big( x^k \big)$ and ...

**9**

votes

**2**answers

552 views

### Consecutive numbers with mutually distinct exponents in their canonical prime factorization

Is it possible to find 23 consecutive positive integers each of which has mutually distinct exponents in its canonical prime factorization? Such numbers are sequence A130091 in OEIS. 24 such numbers ...

**5**

votes

**1**answer

333 views

### A strange condition on containment of special complex numbers in cyclotomic fields

In a recent theorem we have naturally come across this condition, that seems to be important, but rarely satisfied:
$\sqrt{\frac 1 4 + a^m} \in \mathbb Q(\zeta_m, a)$
where $a\in\mathbb C^*$ and ...

**8**

votes

**2**answers

249 views

### Tauberian theorem with better error term

This is a fairly vague question.
Suppose we have a sequence of positive numbers $(c_n)_n$ and we want to find an asymptotic formula for $S(x) = \sum_{n \leq X} c_n$. In favorable circumstances, ...

**0**

votes

**0**answers

27 views

### Is the minimal solution of a Pell equation a positive integral power of the fundamental unit? [migrated]

Let $k=\mathbb{Q}(\sqrt{d})$ -- $d$ is a positive square-free integer --
be a real quadratic field, and let $\varepsilon_k$ be its fundamental unit.
Let $(x,y)$ be the minimal solution to the Pell ...

**1**

vote

**0**answers

128 views

### Generalization of Little Fermat Theorem for a particular $a$ and perfect shuffles

I'm looking for the smallest $n\in \mathbb{N}$ that solves the following equation:
$$2^n=1 \mod m$$
For an odd $m$. I know that Little Fermat Theorem and Euler Totient give me a solution but they ...

**1**

vote

**0**answers

170 views

### What is the status on questions related to Bhargava's factorial function?

In Manjul Bhargava's The Factorial Function and Generalizations he motivates a new type of factorial $n!_S$ using by generalizing a few theorems like:
For $k, l \in \mathbb{Z}$, we have $k! \times ...

**8**

votes

**6**answers

1k views

### number theory which is close to analysis

I have basic training in Fourier and Harmonic analysis. And wanting to enter and work in area of number theory(and which is of some interest for current researcher) which is close to analysis.
...

**3**

votes

**0**answers

124 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, ...

**18**

votes

**0**answers

483 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

**0**answers

82 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 ...

**-4**

votes

**2**answers

296 views

### Brocard's problem [closed]

According to Brocard's problem
$$x^{2}-1=n!=4!*(5+1)(5+2)...(5+s)$$
here,$(5+1)(5+2)...(5+s)=\mathcal{O}(5^{r}),4!=k$. So,
$$x^{2}-1=k *\mathcal{O}(5^{r})$$
Here, $\mathcal{O}$ is Big O ...

**5**

votes

**0**answers

116 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 ...

**3**

votes

**1**answer

143 views

### On conductors, levels and traces on quaternion algebras

I am currently working on level issues in the division central simple algebra case, say $D$ over a local non-archimedean field $F$ (e.g. $\mathbf{Q}_p$). Let say that $\mathcal{O}_D$ and ...

**2**

votes

**0**answers

156 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

**0**answers

77 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

**1**answer

302 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

**1**answer

78 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 ...

**0**

votes

**1**answer

144 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 ...

**10**

votes

**0**answers

116 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

261 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

**2**answers

349 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 ...

**0**

votes

**0**answers

104 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

118 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

**1**answer

200 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

**2**answers

241 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

**2**answers

623 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

**0**answers

173 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

**4**answers

2k 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

**0**answers

82 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

**1**answer

250 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 ...

**7**

votes

**0**answers

290 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

85 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

**0**answers

102 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 ...

**0**

votes

**0**answers

114 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

**0**answers

107 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

**1**answer

163 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

**1**answer

169 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

**0**answers

97 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 ...

**3**

votes

**1**answer

427 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

**1**answer

322 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

**1**answer

373 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

**2**answers

337 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?

**24**

votes

**0**answers

730 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 ...