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

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-1
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0answers
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
2answers
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
4answers
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
1answer
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
0answers
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
3answers
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
2answers
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
0answers
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
0answers
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
3answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
3answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
3answers
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
1answer
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
1answer
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
1answer
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
3answers
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
0answers
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
0answers
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
0answers
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
2answers
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
0answers
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
2answers
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
3answers
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
1answer
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
1answer
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
0answers
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
2answers
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
6answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
2answers
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
5answers
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. ...