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

learn more… | top users | synonyms (1)

0
votes
1answer
107 views

A question on degree 4 binary forms

Suppose that we have a binary form $f(x,y) \in \mathbb{Z}[x,y]$ of degree 4, and that we explicitly have $$\displaystyle f(x,y) = a_0 x^4 + a_1 x^3 y + a_2 x^2 y^2 + a_3 xy^3 + y^4,$$ so that $(0,1)$ ...
-7
votes
0answers
102 views

Theory of mnemonics [on hold]

Even for the typical most skilled (human) number theorist it is hard to reproduce only the first 10 digits of $\pi$ in moderate speed (without physically reading them off). On the other hand there ...
0
votes
0answers
92 views

q-th powers and roots of polynomials

Let $p,q,r$ be integers with $r\ge2$; let $f$ be a polynomial of the form $f(X) = g((X+1)^r)$, which is not a $q$-th power. Let $\omega$ be a $p$-th root of unity. Show that the polynomial ...
2
votes
0answers
102 views

Rational map and diophantine sets

A subset $A$ of $\mathbb{Q}^m$ is a diophantine set over $\mathbb{Q}$ if there is $P(\vec{a},\vec{x}) \in \mathbb{Q}[a_1,...,a_m,x_1,...,x_n]$ such that $\forall \vec{a} \in \mathbb{Q}^{m}$, ...
1
vote
0answers
57 views

Are all complex zeros of $Li_s(i)\, + \, Li_{1-\overline{s}}\,(-i)$ equal to the $\rho$'s?

Take the well known square relationship for polylogarithms: $$Li_s(z)\, + \, Li_{s}(-z)=2^{1-s}Li_s(z^2)$$ Assume $z=i$: $$Li_s(i)\, + \, Li_{s}(-i)=2^{1-s}Li_s(-1)=-2^{1-s}\,\eta(s)$$ with ...
3
votes
1answer
121 views

Optimal lower bounds for the sum of digits in base $b$

Let $b \geq 2$ be an integer and let $s_b(n)$ be the sum of the digits of the base-$b$ representation of the nonnegative integer $n$ (e.g., $s_{10}(726)=7+2+6$). From the weak law of large numbers, it ...
7
votes
1answer
329 views

A question on BSD conjecture

If $E$ is an elliptic curve over $\mathbb{Q}$, and $K$ is an imaginary quadratic field. If $rank E(K)\leq 1$ and both $E$ and the quadratic twist of $E$ by $K$ satisfy the full BSD conjecture, does ...
2
votes
1answer
135 views

Quotient of Projective line over rationals with an infinite subgroup of PGL(2,Q)

I am looking for references for the following; how to calculate quotient of the projective line over the field of rationals with an infinite subgroup of PGL(2,Q), e.g, of the form $ \left( ...
1
vote
1answer
149 views

Asymptotic formula for the number of ways to write a number as the sum of $k$ triangular numbers

How would one derive an asymptotic formula for the number of representations of a number $n$ as the sum of $k$ numbers of the form $\frac{m(m + 1)}{2}$ I think that one could use the circle method, ...
-1
votes
0answers
64 views

Are (odd) perfect numbers divisible by a repdigit (in another base)? How about by a repunit?

[This question was cross-posted from MSE.] A positive integer $N$ is said to be a perfect number if $$\sigma(N) = 2N,$$ where $\sigma(x)$ is the sum of the divisors of $x$. For example, $6$ is ...
0
votes
0answers
52 views

odd squarefree and squareful neighbors [migrated]

There are squarefree numbers $n=\prod_{i=1}^{k}p_i$ so that $n+2$ is not squarefree (e.g. $115+2=3^2.13$). Are there infinite many such $n$? Are there numbers n with arbitrarely many prime-factors? ...
5
votes
0answers
180 views

Invariant obstructions to gluing Galois representations on elliptic curves

Let $E$ be an elliptic curve over $\overline{\mathbb F_p}$, or another separably closed field of characteristic $p$. Let $K$ be the function field of $E$, and let $K_p$ be the local field at a point. ...
0
votes
0answers
147 views

How to find generators to Mordell weil groups of elliptic curves? [on hold]

I am new to branch of elliptic curve and algebraic number theory .I want to find generators to Mordell Weil group of the Elliptic Curve $y^2=x^3-6321363052$ and class number of $\mathbb ...
8
votes
1answer
471 views

Why is the section conjecture important?

As in the title, I want to know the reason for importance of the section conjecture. Of course, the statement of conjecture is important as itself, even I cannot fully grasp the soul of it. However, ...
0
votes
0answers
93 views

Any result, example or conjecture about the computational complexity of transcendental number that can not be computed by linear time Turing Machine [on hold]

The Hartmanis-Stearns Conjecture is restated by Prof.Lipton as: Suppose that a linear time Turing Machine computes the first $n$ digits of the real number $r$ in base ten. Then, the number is either a ...
13
votes
3answers
388 views

Probing the generalization of the abc conjecture to more than 3 variables

Browkin and Brzezinski, in "Some remarks on the $abc$-conjecture", Math. Comp. 62 (1994), no. 206, 931–939, state the following generalization of the $abc$ conjecture to more than three variables: ...
0
votes
0answers
72 views

Are there any relevance between coefficients of simple continued fraction of quadradic algebraic number and algebraic number with degree $2^n$ [on hold]

Let $\sqrt{c}$ be quadratic algebraic number, We know that $[a_0;a_1,a_2,\dots ]$ the coefficients of simple continued fraction of $\sqrt{c}$ the quadratic algebraic number is periodical. ...
9
votes
0answers
135 views

Cubic fields correspond to $3$-torsion ideals in quadratic fields, or to order $3$ characters of quadratic class groups?

I was watching Dick Gross's laudation for Manjul Bhargava, followed up by one of Bhargava's talks, and I realized I was confused about something. Bhargava says (around 21 minutes) that the orbits of ...
9
votes
2answers
350 views

Elliptic curves with trace of Frobenius values always congruent to 0 modulo 2

Let $E/\mathbb{Q}$ be an elliptic curve and suppose that the trace of Frobenius values are such that $a_{p}(E) \equiv 0 \pmod{2}$ for all odd primes avoiding the conductor. Is it the case that $E$ ...
2
votes
0answers
42 views

Image of the typenorm contains the squares

I am having a look at the paper Explicit CM-theory for level 2-structures on abelian surfaces by Bröker, Gruenewald and Lauter, and there is an argument which I don't understand. The main reason being ...
12
votes
0answers
212 views

Artin L-function and Zeta function of twisted Dirac operator

If one thinks of a Frobenius as an element in the fundamental group of an arithmetic curve and of a Galois representation $\sigma$ as a flat connection on the curve, then the definition of the Artin ...
0
votes
0answers
36 views

Stable analytic manifold under simple action

For an integer $m > 1$, let us define the action $$ f: X_i \to (1+X_i)^{m - 1 } $$ on variables $X_1,\dots,X_N$. Consider the analytic manifold $V(I)$ defined by the ideal $I$ in ...
2
votes
2answers
208 views

Reference for Skinner-Urban on the Iwasawa main conjecture for $GL_2$

Does anyone know the existence of an expository paper or a report discussing the work of Skinner-Urban "The Iwasawa main conjecture for $GL_2$"? I am interested in partucular in the case of elliptic ...
1
vote
1answer
147 views

Quadratic Gauss sums: Explicit determinations?

Can anyone please tell me (give me a reference, preferably) if there is any explicit determination of sums of the form $g(n,\chi):=\sum_{r=1}^{q}\chi(r)e(\frac{rn+r^2}{q})$ where $\chi$ is a Dirichlet ...
2
votes
0answers
45 views

On the uncountability of a subset of U-numbers of type $\leq m$

We say that $\xi\in \mathbb{R}$ is an $m$-ultra number if there exists a sequence $(\alpha_n)_n$ of $m$-degree real algebraic numbers, such that $$ |\xi-\alpha_n|<(\exp^{[3]}(H(\alpha_n)))^{-n},\ ...
3
votes
1answer
137 views

Existence of real modular function with specific behavior as $q\to 0$

I am looking for a real modular function $F(q,\bar{q})$ such that in the limit of small $q,\bar{q}$ it behaves as: $F(q,\bar{q})=(a_0 + a_1 (q + \bar q)+...)\log q \bar q+ (b_0 + b_1 (q + \bar ...
1
vote
1answer
210 views

Is it possible to sum the divergent series with prime coefficients?

It is known that the series $$ P := \sum_{n=1}^{\infty} p_{n} \qquad \text{where } p_{n} \text{ is the n'th prime} $$ cannot be summed by means of (prime) zeta function regularization. (The result was ...
0
votes
1answer
205 views

A Problem Concerning Odd Perfect Number

Briefly, prove that every odd number having only three distinct prime factors cannot be a perfect number. I know there are results much stronger than the one above, but I am looking for an answer ...
2
votes
0answers
82 views

What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers $\sqrt{A}$ and the integer $A$

What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers and the integer As we know,$\sqrt{2} = [1;2,2,2,2,…]$; while $\sqrt{14}= ...
2
votes
2answers
359 views

Exponential Sum Bound

In http://131.220.77.52/files/preprints/diophantine/bruedern/wpminicubefinal.pdf, Bruedern and Wooley mention the following fact on the bottom of page 6: Let ...
3
votes
1answer
207 views

Dynamics in the integers - Floor function

Let $\alpha$ be an irrational with $0<\alpha<1$. Consider the function given by \begin{align*} f: &\mathbb{N}\longrightarrow \mathbb{N}\\ &x\longmapsto [ \alpha\cdot x]\end{align*} where ...
0
votes
0answers
203 views

What does a Turing machine compute? [closed]

I suspect that it might be necessary to define for a Turing machine how its inputs and outputs are to be interpreted in order to be able to say e.g. that a Turing machine $T$ computes an arithmetical ...
1
vote
2answers
235 views

Rationale behind an requirement on Turing machines

Hopcroft and Ullman's definition of a Turing machine seems to be standard. This definition defines a Turing machine to be a 7-tupel $T = \langle Q,\Gamma,b,\Sigma,\delta,q_0,F \rangle$ obeying some ...
0
votes
0answers
63 views

Must the radical of polynomial evaluated at integers be small enough at least once?

Basically I am interested if the radical of polynomial evaluated at integers can be small enough at least once. Let $f \in \mathbb{Z}[x], \deg(f)>1$ be squarefree. For integer $a$ and $f(a) \ne 0$ ...
0
votes
1answer
136 views

Composition of a transcendental function with a rational function [closed]

The problem is: let $f: \mathbb{R}\to \mathbb{R}$ be an analytic transcendental function and let $\psi(x)=\frac{x}{2(1+x^2)}$. Is the function $f(\psi(x))$ transcendental?
8
votes
1answer
774 views

What is the arithmetic Nullstellensatz?

The only precise statement (coming from a reliable source) of the "arithmetic Nullstellensatz" I can find is in Gowers's book, stating that two polynomials with integral coefficients have the same ...
16
votes
5answers
804 views
+50

What arrangement of unit cubes minimizes surface area?

For each of these two questions, one can assume that the arrangements are polycubes (for which a definition can be found in the excerpt-image below). Question A. How does one arrange $n$ unit cubes ...
0
votes
1answer
333 views

Counterexample to Pólya's conjecture

It is known that Polya's conjecture is false and the smallest counter-example is about $10^9$. Assuming that we are searching for a counter-example not knowing that it exists. What useful information ...
1
vote
2answers
238 views

Asymptotics on prime divisors

Somewhat inspired from the Zsigmondy's theorem is my question. Suppose Let $a_{1}> a_{2}> \cdots > a_{k}$ be nonzero integers, with $k \geq 2$. Let $a(n) : = a_{1}^{n}+\cdots+a_{k}^{n}$ for ...
-1
votes
0answers
174 views

Where is the Flaw in the Argument? [closed]

Some days ago I have read about the Legendre Symbol. I found that there is no easy method for computation of the symbol. I tried to find out a method for determining the value of ...
2
votes
1answer
193 views

A family Mersenne composite numbers?

I believe that the number $$2^{2^{2t+1}+2t-1}-1$$ is composite for all positive integer $t$. I tested this for many $t$'s, but so far I didn't get a proof. Any idea?
1
vote
0answers
75 views

On the computation of Asai L-function

I want so compute some simple twisted Asai L-function. Let $E/F$ be a quadratic extemsion of number fields and $v$ a finite place of $F$. Let $\chi$ be a unitary automorphic character of ...
2
votes
0answers
148 views

Metric on the set of subsets of the rational primes

Note: this is a revision of an earlier post. It was kindly pointed out that my initial proposed metric was in fact not a metric, so this is a revised version. I was thinking how to say that two sets ...
9
votes
1answer
229 views

Repetend digit graphs for $1/n$ in base $b$

Here is a decimal expansion of $\frac{1}{34}$: $$(1/34)_{10}=0.02941176470588235\overline{2941176470588235}\ldots$$ And here is a graphical representation of the 16-digit "repetend," as a directed ...
6
votes
1answer
222 views

Rational points techniques on curves not using their Jacobian

Let $C/K$ be a curve of genus > 2 over a number field $K$ and suppose there exists a $p \in C(K)$. Then a recurring theme in studying $C(K)$ is using the map $C \to J(C)$ normalized by sending $p$ to ...
5
votes
2answers
181 views

Estimates on derivatives of Bessel function

In Duke, Friedlander and Iwaniec's Erratum on "Bounds for automorphic L–functions. II" They have the following estimates for derivatives of Bessel functions: For $k \geq 2$ \begin{align} & ...
-1
votes
0answers
108 views

Primality matrices [closed]

This question is some kind of a follow-up to my previous thread untitled About Goldbach's conjecture, the content of which follows: 'let's consider a composite natural number $n$ greater or equal ...
10
votes
3answers
1k views

Finding integer points on elliptic curves via divisibility conditions like $(a+b)^2 \mid (2b^3+6ab^2-1)$

Is the following conjecture correct? Conjecture. The divisibility condition $(\alpha+\beta)^2 \mid (2\beta^3+6\alpha\beta^2-1)$ has no solutions in positive integers $1 \le \beta < \alpha < ...
0
votes
0answers
171 views

Which of the Mochizuki's works are the most closely related to elliptic curves?

I'm very much interest about algebraic geometry and number theory along with cryptography, but I have a special interest about the elliptic curves. I have heard a lot of interesting things about ...
-1
votes
0answers
23 views

Proof that $G(3)\le 7$ [migrated]

Let $G(k)$ be the minimal $n$ s.t. every sufficiently large integer is the sum of $n$ nonnegative $k$th powers. Does anybody know where I can find Vaughan's proof that $G(3)\le 7$? I can't find a ...