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

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-2
votes
0answers
46 views

how to calculate the radius of convergence of the p-exponentials of Pulita?

please it is known from the Pulita thesis that the radius of convergence of his pi-exponentials is 1; he used a differential operator which has this pi-exponetial as a solution etc..., but me I ...
4
votes
2answers
591 views

Is every positive integer a sum of at most 4 distinct quarter-squares?

There appears to be no mention in OEIS: Quarter-squares, A002620. Can someone give a proof or reference? Examples: quarter-squares: ${0,1,2,4,6,9,12,16,20,25,30,36,...}$ 2-term sums: ${2+1, 4+1, ...
15
votes
1answer
535 views

Question on the irrationality of $e$

I was surprised that the numbers $\pi$, $\ln{(2)}$, $\zeta{(2)}$, and $\zeta{(3)}$ can be shown to be irrational in what seems to be "three-lined proofs" (as identified here on Overflow: Establishing ...
7
votes
1answer
210 views

how do automorphisms of elliptic curves act on the Tate module?

Let $E/k$ be an elliptic curve over some algebraically closed field $k$ of characteristic $p\ge 0$. It's known that $Aut(E)$ acts faithfully on the Tate module $T_\ell(E)$ ($\ell\ne p$) with ...
5
votes
1answer
137 views

On one class of Somos-like sequences

This question is motivated by integrability of the sequence mistakenly arisen in the question Does this sequence always give an integer? Let $m_1,\ldots, m_{k-1}$ be positive integers and sequence ...
6
votes
0answers
98 views

Integers $d$ for which the Negative Pell equation is soluble for both $d$ and $2d$?

Let $\text{NPE}_d$ denote the negative Pell equation: $$ x^2-dy^2=-1$$ Where $d$ is a given positive nonsquare integer and integer solutions are sought for x and y. we know that (in this paper): ...
1
vote
0answers
257 views

Generalization of proposition of Granville related to abc conjecture

Related to this question. For polynomial $f$, let $rad(f)$ denote the radical of $f$, the product of irreducible factors. Suppose that $G(x,y) \in \mathbb{C}[x,y]$ is homogeneous without any ...
6
votes
2answers
559 views

Counterexample to Proposition of Granville related to abc conjecture

Looks like there is counterexample to Proposition related to abc conjecture. Confusion is likely. From RATIONAL AND INTEGRAL POINTS ON QUADRATIC TWISTS OF A GIVEN HYPERELLIPTIC CURVE, Andrew ...
4
votes
0answers
133 views

Simultaneously using the real and 2adic norms

In the book Modern Computer Arithmetic, there is a section that talks about division with remainder and such in a way that exploits the interplay between the real and 2-adic norms; e.g. the linked-to ...
16
votes
1answer
908 views

Has this strong number theoretic conjecture of Euler been proved, and where could I find such a proof?

Polya cites this work of Euler as an example of a conjecture which Euler considered impossible to doubt, and yet still needing a demonstration. It is on pages 90-98 of "Induction and Analogy in ...
5
votes
1answer
110 views

Bounding a Sum of Adjoint L-Function Values

Fix integers $k\geq2$ and $N>1$, and let $S(k,N)$ denote the normalized new Hecke eigenforms in $S_k(\Gamma_1(N))$. [If it makes my question easier to answer, feel free to replace this with ...
4
votes
3answers
201 views

Is there a generalization of the “characteristic polynomial” to other split/quasi-split algebraic groups?

Let $G = GL_n$ over a field $F$, and let $\gamma \in G(F)$ be a semisimple element. The characteristic polynomial $c_\gamma(t)$ of $\gamma$ encodes a fair bit of information about $\gamma$. ...
2
votes
1answer
69 views

A question about decomposing mod 2 modular forms of level p^2

Fix an odd prime $p$. Each $f \in \mathbb{Z}/2[[x]]$ can be written as $f_{+} + f_{-} + f_0$ where each exponent k of $x$ appearing in $f_{+}$ (resp. $f_{-}$, $f_0$) has Legendre symbol $(k/p)$ equal ...
2
votes
1answer
66 views

On successive minima and basis of a lattice

Let $\Lambda$ be a lattice in $\mathbb{R}^n$. For $\bar{x} \in \mathbb{R}^n$, let $\| \bar{x} \| = max_{1 \leq i \leq n} \{ |x_i| \}$, i.e. the sup norm. Let $\lambda_1, ..., \lambda_n$ be a ...
3
votes
0answers
89 views

Small polynomial roots modulo prime powers

Let $f(x) \in \mathbb{Z}[x]$ have no rational roots, $n>1,q=p^n$ for prime $p$. Let $r$ be the smallest root of $f(x)=0$ modulo $q$. Smallest mean smallest absolute value when lifted to the ...
-2
votes
1answer
113 views

On odd perfect numbers $N$ given in the Eulerian form $N = {q^k}{n^2}$, Part II

I posted this question on MSE two days ago, but did not receive any responses. I have cross-posted it on MO, hoping it gets more attention here and that it is appropriate for this site. A positive ...
1
vote
1answer
97 views

What is the best known lower bound for: $\max_{2\leq i\leq p-1}(ord_n(i)) $?

Given an in integer $n$, and let $p$ be its smallest prime divisor (you can assume that $p$ is very large ). Let $ord_n(i)$ denotes the order of $i$ as an element of $\Bbb Z_n^*$ the multiplicative ...
11
votes
0answers
188 views

Small quotients of smooth numbers

Assume that $N=2^k$, and let $\{n_1, \dots, n_N\}$ denote the set of square-free positive integers which are generated by the first $k$ primes, sorted in increasing order. Question: what is a good ...
-2
votes
1answer
110 views

examples of non-unique factorisation in cyclotomic fields [closed]

I was looking into cyclotomic extensions of the natural numbers, and I found that extending the naturals with the 23rd root of unity caused the ring to no longer be a UFD. In other words, there now ...
2
votes
0answers
200 views

Quadratic Twists on Elliptic Curves- John Coates [closed]

I was actually facing difficulty in Lemma 2.4 of Section 2 (Generalization of Birch's Lemma).-Quadratic Twists on Elliptic Curves- Proceeding of London Mathematical Society. I don't understand why ...
0
votes
1answer
110 views

Algebraic Hecke characters with a given infinite part

I'm needing to find out if there exists an algebraic Hecke character for a number field F, $\phi: \mathbb{A}_F \rightarrow \mathbb{C}$, for a fixed infinite part $\phi_\infty$ and a fixed component ...
2
votes
1answer
133 views

Estermann-Weil bound for Kloosterman sums

Let $m\geq 2$ be a positive integer and consider the Kloosterman sums $$ \mathrm{Kl}(a,b,m)=\sum_{\substack{1\leq x\leq m\\ \gcd(x,m)=1}}\exp\left(\frac{2\pi i}{m}(ax+b\bar{x})\right), $$ where ...
2
votes
0answers
181 views

Algebraic integer with conjugates on the unit circle

Let $\alpha$ be an algebraic integer on the unit circle in $\mathbb{C}$ such that all the conjugates of $\alpha$ lie on the unit circle. Does it follow that $\alpha$ is a root of unity?
7
votes
1answer
188 views

Dedekind-finite arithmetic vs natural numbers arithmetic

It is known that the Dedekind-finite cardinals are closed under addition and multiplication, so one may do arithmetic in them, as opposed to only natural numbers. How much can those two arithmetics ...
4
votes
2answers
264 views

Find all possible rational values of a parametric quartic such that it is reducible

Description: Given the following parametric quartic polynomial $y^4 - 28 z y^3 - 14 (656 - 328 z + 83 z^2) y^2 + 
4 z (-20464 + 10232 z + 3409 z^2) y + 
91 (62208 - 62208 z + 41504 z^2 - 12976 z^3 ...
2
votes
0answers
123 views

Normal basis in cyclotomic number fields

Let $p$ be an odd prime integer and let $\zeta$ be a primitive $2p$-th root of unity. Does $\alpha=1+\zeta+\zeta^{-1}+\dots+\zeta^{\frac{p-1}{2}}+\zeta^{-\frac{p-1}{2}}$ generates a normal basis of ...
5
votes
0answers
159 views

How can I effectively compute tetration mod a?

Is there some general techique to compute tetration and pentation mod some number? $m \uparrow^2 n\mod a$ and $m \uparrow^3 n\mod a$ I know about Euler's theorem to compute $m \uparrow n\mod a$, ...
0
votes
1answer
91 views

Dirichlet series without order term

is there a name in use for Dirichlet series without the order term, analogously to Laurent or Puiseux polynomials? Is there work known about such expressions? $D(s) = \sum_{0<n<N}a_n/n^s$ The ...
6
votes
4answers
487 views

Some Non-Trivial Algebraic(Rational) Number

Every problem about algebraic-ness (rational-ness) of numbers that I have seen is in one of the below types: The number is algebraic(rational) and proving that it is algebraic(rational) is trivial, ...
19
votes
1answer
729 views

Steinhaus's Easter Egg Problem

The following is the text of Steinhaus's so-called Easter egg problem. According to this article of Roman Duda, this was recorded in the New Scottish Book around Easter 1955 and "Steinhaus offered an ...
5
votes
0answers
76 views

Constants for Rosser's Sieve

I am trying to apply Iwaniec's formulation of Rosser's sieve (here) to obtain nontrivial lower bounds for almost-primes in various sequences. These sequences have sieve dimension 1 (if $g(p)$ is the ...
2
votes
0answers
129 views

Diophantine equations and the numbers $4,7,8$

Consider the diophantine equation $$ x^n+y^n+z^n=k\cdot xyz, $$ where $n,x,y,z$ are positive integers. Now consider $k\in\left\{4,7,8\right\}$. It is well-known or easily provable that for $n=1$ and ...
7
votes
5answers
569 views

Open problems in continued fractions theory

I propose to collect here open problems from the theory of continued fractions. Any types of continued fractions are welcome.
3
votes
1answer
180 views

A question on Pythagorean triples

We say that $(a,b,c) \in \mathbb{N}^3$ is a Pythagorean triple if $a^2 + b^2 = c^2$. Is there a characterization of those Pythagorean triples $(a,b,c)$ for which $ab$ is a square-residue modulo $c^2$? ...
10
votes
3answers
348 views

The diameter of a certain graph on the positive integers

Let $G(n)$ be the graph whose vertices are the positive integers $1,2,3,4, \ldots, n$ two of which are joined by an edge if their sum is a square. Is the diameter of this graph 4 for all sufficiently ...
10
votes
0answers
124 views

Hyperelliptic curves over $\mathbb{Q}$ with a $\mu_p^2$ subgroup in their Jacobian

Given a prime number $p>2$, I'm looking for a smooth projective hyperelliptic curve $C$ defined over $\mathbb{Q}$ whose Jacobian $J(C)$ has a subgroup isomorphic to $\mu_p^2$ as a ...
5
votes
0answers
68 views

A Generalized Wiener-Ikehara Theorem with multiple poles on the line

One version of the Wiener-Ikehara Theorem says that if $$ f(s) = \sum \frac{a(n)}{n^s} $$ is a Dirichlet series with nonnegative coefficients that converges absolutely for $\text{Re}(s) > 1$ and ...
4
votes
1answer
205 views

The horizontal distribution of zeros of $\zeta^\prime(s)$

I have a question about a detail in the proof of Proposition 1.6 in "The horizontal distribution of zeros of $\zeta^\prime(s)$", K. Soundararajan, Duke J. Math. vol. 91 1998. Throughout I will ...
1
vote
3answers
125 views

Decidability of sum of powers exponential diophantine equation

I want to ask about decidability of exponential Diophantine equation: $z_12^{\eta_1} + \ldots + z_n2^{\eta_n} = z$, where $z_i,\eta_i,z \in \mathbb{Z}$, and $\eta_i$ - are variables. Can we find ...
0
votes
3answers
164 views

Counting zero-sum free sequences of a given length in $\mathbb{Z}_n$

Let $n$ and $d$ be positive integers. Define $\alpha_n^d$ to be the number of vectors $(x_1, x_2, \cdots, x_d)$ in $\mathbb{Z}_n^d$ such that given any subset $S$ of $\{ 1, 2, 3, \cdots d\}$, ...
4
votes
1answer
139 views

Subgroups of $Sp_{2g}$ giving rise to Shimura data

Consider the Shimura datum $(GSp_{2g},\mathcal{H}_g)$. Let $G$ be a reductive $\mathbb{Q}$-subgroup of $Sp_{2g}$. I want to know under what condition there exists a point $x\in\mathcal{H}_g$ such that ...
4
votes
1answer
303 views

Counting number of points in a lattice with bounded sup norm

Let $\Lambda$ be a lattice in $\mathbb{R}^n$. For $\bar{x} \in \mathbb{R}^n$, let $\| \bar{x} \| = max_{1 \leq i \leq n} \{ |x_i| \}$, i.e. the sup norm. Let $\lambda_1, ..., \lambda_n$ be a ...
12
votes
1answer
388 views

Natural probability on integers

This is a follow-up to this classical question asked recently here: we know (e.g. using the second Borel-Cantelli Lemma) that no probability measure on $\mathbb{Z}$ has the property that $n\mathbb{Z}$ ...
8
votes
2answers
835 views

divisible by all standard prime numbers

This question is about prime numbers in nonstandard models of Peano Arithmetic. Every such model looks like N+AxZ, where A is a dense linear order without end points. There are many nonstandard ...
4
votes
1answer
269 views

Do we know any bound on $lcm(2^1-1, 2^2-1,…,2^n-1)$?

We know that lcm(1,...n) is approximately $e^n$ and and also we know that $gcd(2^a-1, 2^b-1)=2^{gcd(a,b)}-1$. I wonder if there exists an upperbound/lowerbound/approximation for $lcm(2^1-1, ...
0
votes
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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{ ...