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

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4
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1answer
308 views

An elementary question about Gaussian primes

Suppose we consider primes of the form $p = 1 \text { mod } 4$, so that $p = a^2 + b^2$, $a$ and $b$ being integers. Considering only the first quadrant, all $(a,b)$ pairs will be of the form ...
5
votes
1answer
174 views

Is there $t\in\operatorname{Gal}(\overline{K}/K)$ s.t. $\operatorname{rank}_{\mathbf{Z}_p}((t-1)E_{p^\infty}(\overline{K}))=1$?

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, let $$K:=\varinjlim_{k\in\mathbb{Q}[\mu_{p^\infty}]} \mathbb{Q}\left[\mu_{p^\infty},k^{1/p^\infty}\right]$$ and ...
1
vote
0answers
57 views

Completely incongruent box partitions

Let $B$ be a rectangular box with corners in $\mathbb{Z}^d$ and sides parallel to the axes. A completely incongruent partition of $B$ is a partition into $d$-dimensional boxes, each of whose integer ...
2
votes
0answers
112 views

Gcd commutes with $\Psi$, $\Psi\left( \sum_{i=0}^n a_ix^i\right) = \sum_{i = 0}^n a_ix^{p^i}$? [closed]

Define$$\Psi: \mathbb{F}_p[x] \to \mathbb{F}_p[x],\quad\Psi\left( \sum_{i=0}^n a_ix^i\right) = \sum_{i = 0}^n a_ix^{p^i}.$$Do we have that for nonzero polynomials $F$, $G \in ...
11
votes
1answer
219 views

Unusual digit sets that allow finite expansions for all (positive and negative) integers

Informal introduction (If you don't like informal introductions, please skip to 'Mathematical formulation') Whenever our 'decimal positional system' for writing numbers comes up in conversation, ...
5
votes
1answer
472 views

Elliptic curves and prime numbers

Let $p_n$ be the $n^{th}$ prime number. Suppose $E(F_{p_n})$ denotes an elliptic curve over the Galois field $GF(p_n)$ which is defined by $y^2=x^3+ax+b$. Is the below claim true? For each integer ...
29
votes
1answer
1k views

Proving the irrationality of $\pi e$ and $\pi / e$

Rather than relying on the consequences of Schanuel's conjecture, I set about using the same ideas Apery had used to construct integer arguments converging fast enough to show $\zeta(3)$ is irrational ...
1
vote
1answer
89 views

Exponential sum estimates similar to the one for $\sum_p (\log p) e(p \alpha)$, but for different sequences

Obtaining a non-trivial estimate for $\sum_p (\log p) e(p \alpha)$ over the minor arcs is one of the estimates required for obtaining the ternary Goldbach for $n$ sufficiently large via the circle ...
4
votes
3answers
379 views

Conics, rational points and probability

Given a conic $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ with integers and random coefficients, what is more probable? To find a rational point on the conic or not?
3
votes
1answer
154 views

Nontrivial conditions under which $x+y+z$ divides $1 - xyz$

For any nonzero integers $(x, y, z)$ such that $x+y+z$ divides $1 - xyz$., it is easy to verify that $\gcd(x+y, z) = 1$. But what are the nontrivial conditions such that the divisibility holds ?
14
votes
1answer
539 views

Prove $4\sum_{k=1}^{p-1}\frac{(-1)^k}{k^2}\equiv 3\sum_{k=1}^{p-1}\frac{1}{k^2}\pmod{p^2}$

Wolstenholme's theorem is stated as follows: if $p>3$ is a prime, then \begin{align*} \sum_{k=1}^{p-1}\frac{1}{k}\equiv 0 \pmod{p^2},\\ \sum_{k=1}^{p-1}\frac{1}{k^2} \equiv 0 \pmod{p}. \end{align*} ...
0
votes
0answers
64 views

On a (possible?) equivalence of Bunyakovsky conjecture

Dirichlet's Theorem on primes in arithmetical progressions is equivalent to the following statement: for all integer numbers $a,b$, where $\gcd(a,b)=1$, there exists at least one prime of the form ...
1
vote
2answers
241 views

estimate sum of $\log \log p/p$

It is known that $$\sum_{p\leq x} \frac{\log p}{p}=\log x+c.$$ Are any tight bounds on $$\sum_{p\leq x} \frac{\log \log p}{p}$$ known? I haven't managed to find anything in the literature. Trying to ...
10
votes
1answer
376 views

What are “Artin fractions”?

The German Wikipedia entry for Ernst Witt https://de.wikipedia.org/wiki/Ernst_Witt has a photo of his grave in Hamburg. The bottom part has a visible text "Artin Brueche" (Artin fractions) but the ...
13
votes
1answer
357 views

A combinatorial identity involving generalized harmonic numbers

The $n$-th harmonic number is defined as $$ H_n=\sum_{k=1}^{n}\frac{1}{k}, $$ and the generalized harmonic numbers are defined by $$ H_{n}^{(r)}=\sum_{k=1}^{n}\frac{1}{k^r}. $$ Recently, I have found ...
0
votes
0answers
134 views

The existence of solution for special equation on integer ring

I have a question which belongs to the field of number theory. Can we prove or disprove the following claim: For all prime number $p=24t+1$ and the natural number $n=6t+1$, there is at least, one ...
0
votes
1answer
216 views

4th Order Floretions: Floret's Equation [closed]

Update: I've marked this question as answered. If you are thinking "What the heck are floretions?", go right to the answer provided by the Grinch. I definitely should have added clearer information ...
5
votes
4answers
296 views

Counting refinements of partitions

Let $p$ and $q$ be partitions of $n$. We say $q$ refines $p$ if the parts of $p$ can be subdivided to produce the parts of $q$. For example, $(5,5,1)$ refines $(6,5)$ but not $(7,4)$. $(n)$ refines ...
5
votes
1answer
263 views

Good references for K-theory of modular curves?

The title says it. I am looking for a good exposition on the K-theory of the curves $X_{i}(N)$, $Y_{i}(N)$, where $i\in\{0,1\}$. I have some background in $K$-theory and also some background in ...
0
votes
2answers
432 views

Which even numbers are known to be both prime gaps and the sum of 2 primes?

Goldbach's conjecture asserts that every even integer greater than $3$ is the sum of two primes, while de Polignac's one says every even positive integer is a prime gap infinitely often. My question ...
11
votes
1answer
265 views

Characters of weight 1 cusp forms

Assume that the space of cusp forms of weight 1 $S_1(\Gamma_0(N),\chi)$ is non zero. What can one say about the odd character $\chi$, for instance concerning its order ? Serre tells me that a theorem ...
3
votes
1answer
358 views

Algebraic dynamics in finite fields

What is known about combinatorial structure of the rational maps of degree 2 over finite fields? From some general reasons I think it was studied. For being more specific, consider the field ...
7
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0answers
111 views

Equation which has nontrivial solutions modulo $N$ for every $N \ge 2$ does not have any nontrivial integer solutions

Let $\alpha = \sqrt[3]{2}$ and $K = \textbf{Q}(\alpha)$. I want to show that the equation$$\text{N}_\textbf{Q}^K\left(x + 4y + z\alpha + w\alpha^2\right) - 6(x + y)\left(x^2 + xy + 7y^2\right) = ...
1
vote
0answers
36 views

Checking integer points of given infinity norm on intersection

If we have convex region $\mathscr R$ given as intersection of a convex polytope $\mathscr P$ and an ellisoid $\mathscr E$ in $\ell^2$ norm is there an efficient way to test if there is an integer ...
0
votes
1answer
155 views

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

(I have asked a similar question in MSE around a week ago, but did not receive any responses. I have therefore cross-posted it to this site, hoping to get some answers.) An odd perfect number $N$ is ...
15
votes
1answer
641 views

What's so special about these $17$th deg equations?

While browsing the Database of Number Fields, I came across 17T8. It only had four equations, one of which is, $$\small{x^{17} - 5x^{16} + 40x^{15} - 140x^{14} + 610x^{13} - 1622x^{12} + 4870x^{11} - ...
2
votes
0answers
98 views

Zeros of polynomials modulo non-prime

Suppose I have a set S and I want to find a polynomial p such that $p(s) = 0 \mod n$ if $s \in S$, and that it is non-zero modulo n otherwise. In the literature such an S is sometimes called a root ...
4
votes
2answers
145 views

A toy question on solutions to a sequence of trinomials over $\mathbb{F}_3$

Suppose that $\{a_{i}\}_{i\ge1}$ and $\{b_{i}\}_{i\ge1}$ are sequences of natural numbers such that for each $x\in\overline{\mathbb{F}_{3}}^{*}$ there is a natural number $I_{x}$ satisfying that ...
0
votes
0answers
120 views

Difficulty understanding equivalent statement of Erdős Discrepancy Problem

Recently I watched a famous youtube video of talk given by Terry Tao on Erdős Discrepancy Problem https://www.youtube.com/watch?v=QauoO0j9Y9Y. I never heard of this problem before his announcement of ...
5
votes
1answer
166 views

Applications of the Galois embedding problem

Given a finite Galois extension of number fields $L/K$ with Galois group $G$ and a surjection $E\twoheadrightarrow G$ of finite groups, the Galois embedding problem is the question of whether there ...
19
votes
1answer
1k views

What's special about the circle problem?

Let $K$ be a number field, and let $$\zeta_{K}(s):= \sum_{0 \neq I \text{ ideal of }O_K} \frac{1}{N_{K/\mathbb{Q}}(I)^s} = \sum_{n \ge 1} \frac{a_n}{n^s}$$ be the Dedekind zeta function of $K$. The ...
5
votes
1answer
396 views

$(n+1)!_\mathbb{P}$ and the Euler-Mascheroni constant

I'm studying the following limit $$\lim_{n\to \infty} \frac{1}{n} \ln\left( \frac{(n+1)!_\mathbb{P}}{n^n}\right) $$ where $$(n+1)!_\mathbb{P} = \prod\limits_{p \in \mathbb{P}}^{} ...
9
votes
0answers
121 views

Newly defined $L$-function in terms of $L$-function, does it have any obvious zeros or poles?

Let $K$ be a number field, $Cl(K)$ the ideal class group, $\chi: Cl(K) \to \mathbb{C}^\times$ a homomorphism. If $\mathfrak{a} \subset \mathcal{O}_K$ is any ideal, let $[\mathfrak{a}]$ denote its ...
5
votes
2answers
148 views

Reflex fields of Shimura varieties

I am currently learning the theory of Shimura varieties. Out of curiosity, is it known which number fields can occur as reflex fields? More precisely, can one find, for any number field, a positive ...
13
votes
1answer
477 views

What is known about $\sum_{n \leq x} \mu(n) \varphi(n)$?

Let $\mu(n)$ denote the Möbius function and $\varphi(n)$ the Euler-phi function. What is known about $f(x) = \sum_{n \leq x} \mu(n) \varphi(n)$? For example: Is it known that $f(x)$ grows without ...
2
votes
2answers
226 views

A quadratic Diophantine equation

Is the following statement true? If yes, how can find the solutions? The equation $$3x^2+8xy+7y^2\equiv-1\pmod p$$ has an integral solution for every prime $p>5$.
2
votes
1answer
122 views

Upper and lower bounds on reciprocals of restricted prime products

Let $k$ be fixed and consider the sum $$F(k,n)=\sum_{p_1<p_2<\cdots<p_k\leq n~:~p_1 p_2\cdots p_k\leq n} \frac{1}{p_1 p_2 \cdots p_k}.$$ Are tight upper and lower bounds on $F(k,n)$ known as ...
5
votes
0answers
96 views

Sums of twisted products of Kloosterman Sums

For $m,n,c \in \mathbb{N}$, let $S(m,n;c)$ denote the Kloosterman sum $$ S(m,n;c) := \sum_{\substack{1 \leq a < c \\ \gcd(a,c) = 1}} e \left( \frac{ma + n\overline{a}}{c} \right) $$ where $e(n) = ...
3
votes
1answer
106 views

Torsion point on jacobian of ramified cover

Suppose $C$ is a hyperelliptic curve. Then the set of two-torsion points on the jacobians is generated by the set of difference of Weierstrass points. Suppose $C'$ is another hyperelliptic curve. Is ...
12
votes
3answers
587 views

Rational solutions of the Fermat equation $X^n+Y^n+Z^n=1$

As a generalisation to the equation of Fermat, one can ask for rational solutions of $X^n+Y^n+Z^n=1$ (or almost equivalently integer solutions of $X^n+Y^n+Z^n=T^n$). Contrary to the case of Fermat, ...
13
votes
1answer
372 views

Realization of numbers as a sum of three squares via right-angled tetrahedra

De Gua's theorem is a $3$-dimensional analog of the Pythagorean theorem: The square of the area of the diagonal face of a right-angled tetrahedron is the sum of the squares of the areas of the other ...
5
votes
0answers
123 views

Particular case of the class number formula, Dirichlet characters

Let $\chi$ be a Dirichlet character modulo $4$ such that $\chi(-1) = -1$, and let $\chi'$ be a Dirichlet character modulo $5$ such that $\chi'(-1) = 1$, $\chi'(2) = \chi'(3) = -1$. How do I see the ...
8
votes
2answers
395 views

What do we know about the structure of $J_{0}(N)$ over $\mathbb{Q}[{\mu}_{{p}^{\infty}},{{k}}^{\frac{{1}}{{p}^{n}}}])$?

What is known about the structure of $J_{0}(N)$ over $\mathbb{Q}[\mu_{p^{\infty}}]$? More generally, what do we know about $J_{0}(N)$ over $\mathbb{Q}[\mu_{p^{\infty}},k^{1/p^{n}}]$, where ...
0
votes
0answers
154 views

How to compute this sum over numbers?

When I was doing some task of analytic number theory I was stuck on computing this sum $$S:=\frac{1}{L} \sum_{q \in \mathcal{Q}} \phi(q) \overline{a}^{\frac{1}{2}},$$ where $\overline{a}$ is the ...
8
votes
1answer
454 views

Some Questions on the Collatz conjecture

The set of all positive whole numbers is denoted by $\mathbb{N}_+$. Let $f\colon\ \mathbb{N}_+\to\mathbb{N}_+:n\mapsto \begin{cases}\frac{n}{2}&\text{$n$ even}\\3n+1&\text{$n$ ...
1
vote
0answers
55 views

Does Coppersmith's method always finds non-trivial factor of integers of the form $n=a(2^k b+1)$ assuming $1 < a<2^k b +1$ and $b < n^{1/4-0.05}$?

Got an argument and numeric evidence that pari's implementation of Coppersmith's method finds non trivial factor of integers of certain form under some assumptions very efficiently. Three $5000$ bit ...
16
votes
0answers
290 views

What should motives for $L(E,n)$ look like?

Goncharov and Manin showed in this paper that the zeta values $\zeta(n)$ can be realized as periods of framed mixed Tate motives constructed from moduli spaces $\overline{\mathcal{M}}_{0,n+3}$ of ...
8
votes
0answers
198 views

Generating prime numbers

By a theorem of Mills, 1947, there is a real number $c$ such that for every $n$, $[c^{3^n}]$ is a prime number. Is there a real number $d$ such that $[d^n]$ is prime, for every $n$ ?
2
votes
1answer
172 views

Magic tesseract of order 3 composed of prime numbers

Definition. A magic tesseract is a four-dimensional array, equivalent to the magic cube and magic square of lower dimensions, containing the numbers 1, 2, 3, …, m^4 arranged in such a way that the ...
12
votes
2answers
221 views

Non-congruence normal subgroups of $SL_2(\mathbb{Z}[1/2])$

Let $G=SL_2(\mathbb{Z}[1/2])$, i.e., the modular group (if you wish) over the ring $\mathbb{Z}[1/2]$ consisting of rationals whose denominators are powers of $2$. Unlike $SL_2(\mathbb{R})$, $G$ is ...