# Tagged Questions

**-1**

votes

**0**answers

177 views

### Is there any significance in Heegner numbers (or class number 1) representation symmetry?

$\mathrm{A003173}(n) = 1+((1 + \sqrt{3})^{n-1} - (1 - \sqrt{3})^{n-1})/(2\sqrt{3})$
for n = 1,2,3,4.
$\mathrm{A003173}(n) = 19+24((1 + \sqrt{3})^{n-6} - (1 - \sqrt{3})^{n-6)})/(2\sqrt{3})$
for n = ...

**5**

votes

**2**answers

836 views

### Riemann hypothesis and Kakeya needle problem

The question may be a bit vague. I noticed an analogy that both Riemann hypothesis and Kakeya needle problem has been proved in finite fields. Can somebody shed light on why finite field analogues are ...

**5**

votes

**4**answers

358 views

### How to get a good upper bound on $\sum_{1 \lt d, d|n} \phi(d)/\log d$?

I'm actually interested in a slightly smaller quantity, but I'm willing to accept the following simplification, especially if there are small error terms.
Let's start with $ n \gt 1$, Euler's totient ...

**4**

votes

**1**answer

79 views

### Weight filtration on certain Galois representations

Let $G$ be the absolute Galois group of a number field $K$. Let $\ell$ be a prime number. There are representations $\mathbb{Z}_\ell(n)$ of $G$ on the group of $\ell$-adic integers given by the ...

**0**

votes

**0**answers

77 views

### Are there any ways we can determine whether the $\Xi_x$-classes of natural numbers upto $\frac{1}{2}p^2_x -2$ exvert all non-trivial $\Xi_x$-classes? [on hold]

This question follows from the information provided below.
Are there any ways we can determine whether the $\Xi_x$-classes of natural numbers up to $\frac{1}{2}p^2_x -2$ exvert all non-trivial ...

**14**

votes

**3**answers

482 views

### Are there open problems for primes which are known for probable primes?

Define "probable prime" (PP) to be natural $n>1$ satisfying $2^{n-1} \equiv 1 \pmod{n}$ or $n=2$.
Probable primes are the union of the primes and base two pseudoprimes.
This definition is much ...

**21**

votes

**2**answers

481 views

### Elementary congruences and L-functions

In a recent article, Emmanuel Lecouturier proves a generalization of the following surprising result: for a Mersenne prime $N = 2^p - 1 \ge 31$, the element
$$ S = \prod_{k=1}^{\frac{N-1}2} k^k $$
...

**17**

votes

**0**answers

1k views

### Is formula valid for relating $\pi$ with ALL of its OEIS A002485(n)/A002486(n) convergents?

Could anyone try to prove that the below conjectured formula is valid for relating $\pi$ with ALL of its convergents - those, which are described in OEIS via A002485(n)/A002486(n)?
$$(-1)^n\cdot(\pi ...

**0**

votes

**0**answers

182 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 ...

**5**

votes

**3**answers

228 views

### Asymptotics for the number of abelian groups of order at most $x.$

The number of abelian groups of order $n$ (call it $a(n)$ is a studied subject (see http://oeis.org/A000688), but I can't seem to find any asymptotic results. Obviously, there is no asymptotic for ...

**4**

votes

**1**answer

218 views

### Can we always attain another prime via inserting digits between the digits of a fixed prime?

The sequence OEIS A080437 is
For n > 10, let m = n-th prime. If m is a k-digit prime then a(n) = smallest prime obtained by inserting digits between every pair of digits of m.
I don't see why ...

**0**

votes

**1**answer

258 views

### If $N = qn^2$ is an odd perfect number, is it possible to have $q + 1 = \sigma(n)$?

The title says it all.
Question
If $N = qn^2$ is an odd perfect number with Euler prime $q$, is it possible to have $q + 1 = \sigma(n)$?
Heuristic
From the Descartes spoof, with quasi-Euler ...

**4**

votes

**1**answer

349 views

### natural radical and an algebraic expression in $\pi$ and/or $e$

Let $\ \mathbb N:= \{1\ 2\ \ldots\}\ $ be the set of natural numbers. Let $\ \mathbb P:=\{2\,\ 3\,\ 5\,\ 7\,\ 11\,\ \ldots\}\ $ be the set of primes. Then natural radical $\ rad(n)\ $ is
$$ rad(n)\ ...

**3**

votes

**1**answer

391 views

### Use of infinitude of primes in the Green-Tao theorem [on hold]

In a video I watched last night on nuking mathematical mosquitos, Matt Parker gave the following proof of the infinitude of primes: suppose there are finitely many primes. The Green-Tao theorem says ...

**3**

votes

**1**answer

157 views

### A decreasing sequence involving the divisor function?

Define $N_k \geq 6$ to be the $k-th$ primorial number and let $\sigma(n)$ be the divisor function.
It seems that $u_k = \dfrac{\sigma(N_k)}{N_k \log\log N_k}$ is a decreasing function ?
By ...

**0**

votes

**0**answers

18 views

### Standard term for parametrisation where heights of parameters and values are correlated

Suppose an algebraic variety over Q, or subvariety of one, has a parametrization, also over Q.
Clearly an infinite number of birationally equivalent parametrisations can be obtained from this. But ...

**4**

votes

**1**answer

86 views

### Sufficient conditions for $\sum_{n \ge 1} a_n e^{-(a_1+\cdots+a_n) s} \sim \frac{1}{s}$ as $s \to 0^+$

Let $(a_n)_{n \ge 1}$ be a sequence of non-negative real numbers such that $\sum_{n \ge 1} a_n = \infty$, and set $\lambda_n := a_1 + \cdots + a_n$ for each $n$. Then the (generalized Dirichlet) ...

**6**

votes

**1**answer

168 views

### Property of Dirichlet character

Let $\chi_0$ be the unique Dirichlet character $\text{mod }1$ (i.e. $\chi_0(n) = 1$ for all $n$), $\zeta_p$ be a primitive $p$th root of $1$, and for any $a \in \mathbb{F}_p^\times$ and any Dirichlet ...

**2**

votes

**3**answers

195 views

### Fricke Klein method for isotropic ternary quadratic forms

Preface: the most natural way to take one isotropic vector for an indefinite quadratic form and find others is to use stereographic projection. This gives a parametrization in the same $n$ variables ...

**2**

votes

**0**answers

80 views

### When is $(1^2+1)(2^2+1)\dots (n^2+1)$ a perfect square? [duplicate]

Find all such $n$. Natural guess is that $n=3$ is the only solution.
It is natural to try something like Bertrand's postulate for Gaussian integers with imaginary part 1, but what is known?

**2**

votes

**0**answers

99 views

### Exponential analogue of formal connections

Let $F=\mathbb{C}((t))$. Let $G=GL_n$. Then $G(F)$ acts on $\mathfrak{g}(F)$ by gauge transformation:
$$
g.x:=gxg^{-1} + \dot{g}g^{-1},\quad \quad g\in G(F), \quad x\in \mathfrak{g}(F).
$$
Here, ...

**3**

votes

**1**answer

310 views

### Primary structures in $\mathbb Q$

I'll formulate a topic restricted here to the positive rational
numbers $\ \mathbb Q_{_{>0}},\ $, then will pose a question (Q2) plus some related, to which I don't know the answers nor reference. ...

**9**

votes

**1**answer

428 views

### A divisor sum congruence for 8n+6

Letting $d(m)$ be the number of divisors of $m$, is it the case that for $m=8n+6$,
$$ d(m) \equiv \sum_{k=1}^{m-1} d(k) d(m-k) \pmod{8}\ ?$$
It's easy to show that both sides are 0 mod 4: the left ...

**6**

votes

**1**answer

204 views

### Up to $2000$, $A145722(n-1) \equiv \sigma(4n-3) \pmod{5}$

A145722 is
Expansion of f(q) * f(q^5) / phi(-q^2)^2 in powers of q where f(), phi() are Ramanujan theta functions.
Using the pari program and offset 0, up to ...

**8**

votes

**1**answer

708 views

### Up to $10^6$: $\sigma(8n+1) \mod 4 = OEIS A001935(n) \mod 4$ (Number of partitions with no even part repeated )

Up to $10^6$:
$\sigma(8n+1) \mod 4 = OEIS A001935(n) \mod 4$
A001935 Number of partitions with no even part repeated
Is this true in general?
It would mean relation between restricted partitions ...

**6**

votes

**3**answers

193 views

### Uniform bounds on the number of integer points on a family of elliptic curves

Let $P(x,y)$ be a binary cubic polynomial with integer coefficient. Let $n$ be an integer. Suppose the (complex) curve $P(x,y)=n$ is nonsingular, so is an elliptic curve. Is there any bound on the ...

**11**

votes

**1**answer

566 views

### Why does $H^i(X_{ét},\mathbb{Q}_p)$ have a Hodge-Tate structure?

Let $X$ be a variety over a $p$-adic field $K$.
Is there a simple or intuitive explanation of why the $G_K$ representation $H^i(X_{ét},\mathbb{Q}_p)$ is Hodge-Tate? More precisely, why do the powers ...

**3**

votes

**1**answer

235 views

### Convergence of a triple sum involving the imaginary part of the Riemann zeta function's non trivial zeros

Let $N>0$ an integer, $k>0$ a real parameter and let $\rho = \beta +i \gamma$ a non trivial zero of the Riemann zeta function. For a work I need to find the best possible $k$ such that ...

**9**

votes

**1**answer

425 views

### Who was/were the first to note that if $\sum_{x \in X} \frac{1}{x} < \infty$ then the natural density of $X$ is zero?

It is a result of folklore that the natural density of a set $X$ of positive integers such that $\sum_{x \in X} \frac{1}{x} < \infty$ is zero. This is reproved, e.g., in T. Šalát's paper: ...

**15**

votes

**3**answers

918 views

### What is a sieve and why are sieves useful?

I have been trying to understand what is exactly a sieve and why sieves are useful.
I have read Wikipedia articles about sieve theory but they don't provide a definition of what is a sieve or why they ...

**1**

vote

**0**answers

35 views

### On modulus of powers

Given $n\in\Bbb N$ is there an $a_n\in\Bbb N$ such that for every $a>a_n$ there are $b,c\in\Bbb N$ such that $b^i\bmod a,c^i\bmod a\in(\sqrt a,\sqrt a\log a)$ for every $i\in\{1,\dots,n\}$? If so ...

**7**

votes

**0**answers

97 views

### The non-abelian Gauss-Manin connection; non-abelian M_dR; a Grothendieck lemma for cyrstals

I'm interested in understanding the non-abelian Gauss-Manin connection on Carlos Simpson's relative de Rham Moduli space $M_{dR}(X/S,n)$ for a smooth projective morphism of schemes $X/S$. The scheme ...

**1**

vote

**0**answers

205 views

### Is the rank of the elliptic curve x^3 + y^3 = a(n), where a(n) is the n-th cubefree taxicab number, known?

Is the rank of the elliptic curve x^3 + y^3 = a(n) over the rationals, where a(n) is the n-th cubefree taxicab number A080642(n) in the OEIS, known?

**9**

votes

**2**answers

729 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 ...

**9**

votes

**0**answers

572 views

### Is there a proof that OEIS-A002387 is $[ e^{n-\gamma} ]$?

Based on the comments on OEIS-A002387:
$a_{n}$ = 1, 2, 4, 11, 31, 83, 227, 616,...
it is likely, that the sequence $a_{n}$ coincides with $[ e^{n-\gamma} ]$ ,
where $\gamma$ is the Euler-Mascheroni ...

**3**

votes

**2**answers

387 views

### Are all counterexamples of OEIS A226181 both Poulet numbers and Proth numbers?

OEIS A226181:
3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 47, 53, 59, 61, 67, 71, 73, 79,
83, 89, 97, 101, 103, 107, 113, 131, 137, 139, 149, 163, ...
Primes $p$ ...

**5**

votes

**3**answers

506 views

### If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal

On p. 76 of the 1996 edition of Serre's A Course in Arithmetic, one reads the following (inline) remark:
One can prove that, if $A$ has natural density $k$, the analytic density of $A$ exists and ...

**4**

votes

**1**answer

203 views

### Uniformizer for splitting field of p^{1/p^n} over p-adics

Does anyone know an explicit uniformizer for $\mathbf{Q}_p(\zeta_{p^n}, p^{\frac{1}{p^n}}) / \mathbf{Q}_p$? I was reading the question "adding an n-th root to Q_p" where dke mentions this question but ...

**2**

votes

**1**answer

288 views

### On the Diophantine equation $x^2 = y^p + 2^{r}z^p$ where $p\geq 7$ is an odd prime and $r \geq 2$

It is known that the only nonzero pairwise coprime integer solutions to the above Diophantine equation are for $r=3$, for which $(x, y, z) = (3,1,1)$ and $(-3,1, 1)$. (Cohen, Number Theory Volume 2: ...

**-4**

votes

**0**answers

107 views

### I am looking for a general solution for when $n$ and a rational function $f (n)$ are both integers [closed]

I am looking for a general solution for when $n$ and a rational function $f \left({n}\right)$ are both integers. One example is below.
This seams simple, how to prove that the only integer solutions ...

**23**

votes

**2**answers

801 views

### $\zeta(n)$ as a mixed Tate motive

I am trying to understand why there exists, for each $n \geq 2$, a mixed Tate motive $M$ over $\mathbb{Q}$ such that
$M \in Ext^1_{MT(\mathbb{Q})}(\mathbb{Q}(0), \mathbb{Q}(n))$
and $\zeta(n)$, ...

**12**

votes

**6**answers

2k views

### English reference for a result of Kronecker?

Kronecker's paper Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten apparently proves the following result that I'd like to reference:
Let $f$ be a monic polynomial with integer ...

**17**

votes

**1**answer

341 views

### How big are the prime factors of $2^kp - 1$?

I have already asked this question here. No answers despite the bounty (which has now ended)
Let $p$ be a prime number, $p > 3$.
Does there always exist $k \in \mathbb N_{\ge 1}$ such that the ...

**1**

vote

**0**answers

54 views

### Sampling efficiently conditioned on linear constraints modulo both $\mathbb{F}_p$ and $\mathbb{F}_2$

Given a prime $p$ and positive integer $t \ll \log p$ (say $t = \sqrt{\log p}$), is there an algorithm that is polynomial time in $\log p$ to sample uniform $X, Y \in \mathbb{F}_p$ conditioned on the ...

**5**

votes

**1**answer

301 views

### Local Langlands correspondence and Galois equivariance

The local Langlands correspondence $\text{rec}$ for $\text{GL}_{n}$ itself is not Galois equivariant (i.e. invariant under automorphisms of its field of definition) but rather its twist by ...

**21**

votes

**0**answers

375 views

+50

### Is the Flajolet-Martin constant irrational? Is it transcendental?

Facebook has a new tool to estimate the average path length between you and any other person on Facebook. An interesting aspect of their method is the use of the Flajolet-Martin algorithm.
In the ...

**0**

votes

**0**answers

35 views

### If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, is $n^2$ solitary?

(Note: A similar question was asked in MSE two months ago.)
Let $\sigma(x)$ be the sum of the divisors of the natural number $x$, and denote the abundancy index $\sigma(x)/x$ by $I(x)$.
Here is my ...

**-1**

votes

**0**answers

151 views

### On even almost perfect numbers other than the powers of two, as compared to odd perfect numbers given in Eulerian form

This question has been cross-posted from MSE.
Let $\sigma(x)$ be the sum of the divisors of $x$. We say that $X$ is almost perfect if $\sigma(X) = 2X - 1$.
Antalan and Tagle (in a 2004 preprint ...

**8**

votes

**2**answers

310 views

### What's in the genus of the cubic lattice?

I'll write $\mathbf{Z}^n$ for the integral quadratic form $x_1^2 + \cdots + x_n^2$. For which values of $n$ is $\mathbf{Z}^n$ unique in its genus, i.e. isolated in Kneser's graph? In particular can ...

**34**

votes

**5**answers

2k views

### The sequence $a_{n+1}=\left\lceil \frac{-1+\sqrt{5}}{2}a_{n}-a_{n-1} \right\rceil$ is periodic

Let $(a_{n})_{n \ge 1}$ be a sequence of integers such that for all $n \ge 2$:
$0\le a_{n-1}+\frac{1-\sqrt{5}}{2}a_{n}+a_{n+1} <1$.
Prove that the sequence $(a_{n})$ is periodic.
...