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

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0
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29 views

Find all integers x, y, and z such that 1/x + 1/y = 1/z [migrated]

Characterize all positive integers x, y, and z such that $\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$. For example, $\frac{1}{x+1} + \frac{1}{x(x+1)} = \frac{1}{x}$
-1
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0answers
134 views

About a problem requiring long calculations [on hold]

Given six points: $$P₁=(129/256,65/256)=(x₁,y₁)$$ $$P₂=(132097/262144,66561/262144)=(x₂,y₂)$$ $$P₃=(2164277249/4294967296,1090535425/4294967296)=(x₃,y₃)$$ $$P₄=(δ_{q-1}-2α_{q-1},β_{q-1})=(x₄,y₄)$$ ...
1
vote
0answers
77 views

Averages of $L(s,\chi)$

Let $(\frac{m}{n})$ denote the usual quadratic Jacobi symbol. What is the abscissa of convergence of the double Dirichlet series ? $$ \sum_{\substack{m,n \in \mathbb{N} \\ \gcd(m,n)=1 \\m,n\equiv 1 ...
1
vote
1answer
79 views

Q(\sqrt{-l_0}) satisfies Heegner hypothesis for an Elliptic curve of conductor C implies C is a square

Trying to understand the proof of Corollary 2.3 in the following paper, http://arxiv.org/pdf/1312.3884.pdf I would like to be able to justify that the root number of the quadratic twist ...
0
votes
0answers
66 views

Integer points on Elliptic Curves [on hold]

It's easy to prove that equation y^2=x(x-a)(x-b) with a and b integer has integer solution, other than (0,0), (a,0) and (b,0), if a and b jointly admits the representation a=(r-s)r (1) b=(r-t)t ...
7
votes
0answers
113 views

Geometric interpretation of Cusps for general groups?

Let $\mathrm{G}$ be a reductive group over a number field $F$, but for simplicity we can think about $\mathrm{G}=\mathrm{GL_n}$ for $n>2$ and $F =\mathbb{Q}$. Then for an automorphic form, ...
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0answers
35 views

Quadratic residue [on hold]

P is prime. Prove -1 is a quadratic residue mod p if and only if either p = 2 or else p =1(mod4) I tried using quadratic reciprocity proofs to solve, but am unable to discover the answer.
3
votes
0answers
179 views

The $\ell = p$ case of Ihara's lemma

Let $N \ge 1$ and let $\ell$ and $p$ be primes not dividing $N$. The classical Ihara lemma says that if $Y_1(N, \ell)$ is the modular curve attached to the subgroup $\Gamma_1(N) \cap \Gamma_0(\ell)$, ...
2
votes
0answers
110 views

irrationality of Dedekind zeta values

For Riemann's zeta function, one knows that $\zeta(2n)$ is irrational (because $\pi$ is) $\zeta(3)$ is irrational (proved by Apéry) and a few other results like "there are infinitely many ...
1
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0answers
46 views

Euler series with milder divergence

Theorema 19 in Euler's memoir "Variae observationes circa series infinitas" says The sum of the reciprocals of the prime numbers is infinitely great but is infinitely times less than the sum of the ...
4
votes
2answers
171 views

Bound on a scaled sum of the Liouville function

Terence Tao has shown see his blog post that $$\left| \sum_{n\leq x} \frac{\mu(n)}{n} \right|\leq 1,$$ for $x$ a positive real number, where $\mu(n)$ is the Möbius function. Let $\lambda(n)$ denote ...
7
votes
2answers
466 views

Most dense subset of numbers that avoids arbitrarily long arithmetic progressions

The famous Green-Tao theorem says that there exist arbitrarily long sequences of primes in arithmetic progression. I am wondering: How dense can a subset $S \subset \mathbb{N}$ be and still avoid ...
0
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0answers
106 views

injective homogeneous polynomial functions $p(x,y) \in \mathbb{Z}[x,y]:{\mathbb{N}}^2 \to \mathbb{N}$

Related to this question Coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$ I have the following question: What is the set of homogeneous polynomials ...
1
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0answers
119 views

Coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$

I want to prove that $\forall n \in \mathbb{N}$ at least one of the Fourier coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$ is striclty greater than ...
4
votes
2answers
272 views

What is the Hausdorff dimension of this fractal?

Let $\sum_{i=h}^\infty d_i/b^i $ be the base $b$ representation of $x \geq 0,$ where $b>1$ and the $d_i$ are uniquely determined by the greedy algorithm. For fixed $c>1,$ let $f(x)= ...
1
vote
0answers
78 views

Best constant for Maier's theorem?

Maier proved that, for fixed $\lambda>1,$ $$ \limsup_{x\to\infty}\frac{\pi(x+\log^\lambda x)-\pi(x)}{\log^{\lambda-1}x}>1 $$ and in particular $$ \limsup_{x\to\infty}\frac{\pi(x+\log^\lambda ...
10
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0answers
239 views

Why would the roots of the generating functions of the number of k-almost primes less than x have negative real parts?

Specifically, I find it appealing to count only squarefree numbers having $k$ prime factors, so I define $$\pi_k(x)=\#\{n\leq x: \omega(n)=k;\mu(n)\neq0 \}$$ and consider the generating functions ...
1
vote
2answers
303 views

Examples that the Fermat-Catalan conjecture does not cover

The Fermat-Catalan conjecture states that there are only finitely many sex-tuples $(a, b, c, d, e, f)$ of positive integers such that (1) $a^d + b^e = c^f$, (2) $\gcd(a, b, c) =1$, (3) ...
2
votes
1answer
115 views

Irreducibility of Faulhaber-like Polynomials over $\mathbb Q[x]$

Motivation: Inspired by the famous Faulhaber polynomials $F_k(N)=\displaystyle\sum_{n=0}^Nn^k,$ I decided to study their alternating versions, $\Phi_k(M)=\displaystyle\sum_{n=0}^M(-1)^nn^k$. For ...
4
votes
1answer
240 views

Square-free grows as $6n/\pi^2$: $k$-th free?

The asymptotic number of square-free numbers $\le n$ is $Q(n) = 6n/\pi^2 + O(\sqrt{n})$. Because $\zeta(2)=\pi^2/6$, $Q(n) \approx n/\zeta(2)$. OEIS A004709 says that cube-free numbers have ...
3
votes
1answer
252 views

Invertibility of a matrix whose entries are certain binomial coefficients

Let $l$ be a positive integer. Does the matrix $$ M_l \ := \ \left( \binom{l-(2p+1)}{j} \right)_{0\leq p,j \leq[(l-1)/2]} $$ have nonzero determinant?
1
vote
1answer
205 views

Is there a non-tempered representation of U(2)?

I am wondering why the first well known example of irreducible admissible representation of $p$-adic group $U(n)$ should be $U(3)$. Because, Gelbart and Rogawski suggested the non-tempered ...
0
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0answers
65 views

Regular Magic Squares [closed]

I am currently studying magic squares and ran into a bit of trouble. Basically it deals with something I came up with, a regular square. Below are the conditions of a regular square. We can say that ...
9
votes
2answers
781 views

What is $\sum_{i=0}^{n}\binom{n}{i}^3$?

We know that $$\sum_{i=0}^{n}\binom{n}{i}=2^n$$ and that $$\sum_{i=0}^{n}\binom{n}{i}^2= \binom{2n}{n}$$ what about $$\sum_{i=0}^{n}\binom{n}{i}^3$$ ?
6
votes
1answer
142 views

When are toral orbits in buildings the difference of fixed-sets?

Let $L$ be a $p$-adic field, let $G$ be a reductive group over $L$ (I'm even okay assuming semisimplicity for now). Let $T$ be a maximal torus of $G$. Let $B$ be the building for $G(L)$. (Edit 1: ...
4
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1answer
192 views

Counting couples of square-free polynomials over finite fields

I have a curve defined by the following equations over the finite field $\mathbb{F}_q$ with $q=p^r$ with $p \geq 3$: $$C_{h_1,h_2}:\begin{cases} y_1^2=h_1(t) & \\ y_2^2=h_2(t) ...
16
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4answers
2k views

A question on Collatz's conjecture

Let $C$ : ${\mathbb N}\longrightarrow {\mathbb N}$ be Collatz's map defined by $C(n) = 3n+1$ if $n$ is odd, and $C(n)=n/2$ if $n$ is even. Then according to Collatz's conjecture, we should have $C^k ...
8
votes
2answers
682 views

What is known about primes of the form x^2-2y^2?

David Cox's book Primes of The Form: $x^2+ny^2$ does a great job proving and motivating a lot of results for $n>0$. I was unable to find anything for negative numbers, let alone the case I am ...
8
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0answers
210 views

Geometric vs combinatorial motives over Spec Z

Consider the category of reduced schemes of finite type over $\mathbb{Z}$. Take the Grothendieck group of this category, i.e. the free abelian group on isomorphism classes, modulo the usual "syzygy" ...
2
votes
1answer
259 views

$n^3 | \sum_{i=1}^{n-1}\binom{n}{i}^2$ => $n | \sum_{i=1}^{n-1}\binom{n}{i}$?

For $n\in \mathbf{N}$ is $$n^3 \text{ divides } \sum_{i=1}^{n-1}\binom{n}{i}^2=\binom{n}{1}^2+\cdots +\binom{n}{n-1}^2$$ impling $$n \text{ divides } ...
1
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0answers
66 views

Spectrum of primitive nonnegative integer matrices

Let $P(X) = a_nX^n + \cdots + a_1X + a_0$ with $a_i \in \mathbb Z$. Question 1. Is there an efficient criterion on the $a_i$ to decide if there exists a primitive nonnegative integer matrix with ...
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0answers
265 views

Is there a polynomial that generates only primes or semi-primes? [migrated]

I know that no non-constant polynomial function $P(n)$ with integer coefficients exists that evaluates to a prime for every integer value of $n$. My question is - does there exist a non-constant ...
21
votes
2answers
3k views

A possibly surprising appearance of Fibonacci numbers

Let $H(n) = 1/1 + 1/2 + \dotsb + 1/n,$ and for $i \leq j,$ let $a_1$ be the least $k$ such that $$H(k) > 2H(j) - H(i),$$ let $a_2$ be the least k such that $$H(k) > 2H(a_1) - H(j),$$ and ...
2
votes
0answers
93 views

Are all zeros of $\xi(a\,s) \pm \xi\left(a\,(1-s)\right)$ on the critical line for $\forall a \in \mathbb{R}/0$?

This question expands on this one and seems to have a stronger result. Take the Riemann $\xi$-function $\xi(s) =\frac12 s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s)$. We ...
4
votes
0answers
276 views

Conjectures on perfect squares

There exist infinitely many sets of three positive integers $(a,b,c)$ where $a\not=b,b\not=c$ and $c\not=a$ such that each of $$ab-1,\ bc-1,\ ca-1,\ ab-a-b+c,\ bc-b-c+a,\ ca-c-a+b$$ is a perfect ...
4
votes
1answer
557 views

How good can we approximate with algebraic curve the egg shaped vanishing of $\Re \zeta(s)$ near the origin?

Related to this question where degree $2$ algebraic curve is good approximation to vanishing of the real part of expression involving zeta. Near the origin, $\Re \zeta(s)$ vanishes in egg shaped ...
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0answers
95 views

How comes vanishing of the real part of function involving zeta is very well approximated by algebraic curve?

In this question Agno asked about the zeros of $\zeta\left(\frac{s}{a}\right) \pm \zeta\left(\frac{1-s}{a}\right)$. I fixed $a=2$ and the minus sign and defined: $$ f(s)=\Re \left( ...
0
votes
0answers
132 views

Does the Euler product converge at $s=1$ for the Dirichlet $L$ function?

For the Riemann Zeta function, the Euler product converges on $\{Re(s)=1\}$ except at $s=1$.The zeta series diverges everywhere on $\{Re(s)=1\}$. But the $L$ series converges on $\{Re(s)>0\}$. What ...
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0answers
227 views

Is the absolute Galois group of the rationals Hopfian?

Is every continuous epimorphism from the absolute Galois group of $\mathbb{Q}$ to itself injective?
7
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1answer
230 views

Set of integers having finite intersection with the image of any polynomial of degree $\geq 2$

Is there a set $A$ of positive integers such that $\sum_{n \in A} \frac{1}{n} = \infty$, and there is no polynomial $f \in \mathbb{Z}[x]$ of degree at least $2$ which takes infinitely many values in ...
2
votes
0answers
50 views

Are the complex zeros of $\zeta\left(\frac{s}{a}\right) \pm \zeta\left(\frac{1-s}{a}\right)$ all on the critical line for $a \lt 0, a \ge 1$?

With $s \in \mathbb{C}, a \in \mathbb{R}$, numerical evidence strongly suggests that the complex zeros in the critical strip of: $$\zeta\left(\frac{s}{a}\right) \pm ...
3
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0answers
73 views

When do two lattices have the same stabilizer in the diagonal torus?

This is moved from MSE, where I asked and didn't receive an answer (see http://math.stackexchange.com/questions/1145151/lattices-in-mathbbq-pn-with-the-same-stabilizer) Let $T$ be the diagonal torus ...
16
votes
1answer
710 views

Primes that are sums of two squares with constraints on the squares

It is well known that there are infinitely many primes of the form $a^2+b^2$ (namely all primes congruent to $1$ modulo $4$). On the other hand, Euler raised the problem as to whether there are ...
1
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0answers
87 views

Limit of a simple function including a zero of the Riemann Zeta function

Lets consider : $$F(x)= \sum_{n\in\mathbb{N}} n^{-s_0} e^{2i\pi nx}$$ This function is well defined for $x>0$ (Abel summation formula proves it) and I would like to show that if $s_0$ is a zero ...
6
votes
0answers
160 views

P-depletion of Siegel modular forms

Let $F$ be a cuspidal Siegel modular form of genus 2 (of parallel weight $(k, k)$, and level some congruence subgroup $\Gamma \subseteq Sp_4(\mathbf{Z})$ of level $N$). Then $F$ has a series ...
1
vote
1answer
120 views

Infinite sum of asymptotic expansions

I have a question about an infinite sum of asymptotic expansions: Assume that $f_k(x)\sim a_{0k}+\dfrac{a_{1k}}{x}+\dfrac{a_{2k}}{x^2}+\cdots$ with $a_{0k}\leq \dfrac{1}{k^2}$, $a_{1k}\leq ...
0
votes
0answers
61 views

Is it possible the division polynomials evaluated at fixed point to be perfect powers unbounded number of times?

Let $E$ be elliptic curve over the rationals and $P=(X_P,Y_P)$ point on $E$. $\psi_n$ are the division polynomials. Define $a_n=\psi_n(X_P,Y_P)$. Is it possible $a_n$ to be perfect power ...
1
vote
1answer
161 views

Reference request: Research done on whether the Euler prime can be the largest factor of an odd perfect number

(Note: This was cross-posted from MSE.) I posted the following reference request in MSE three (3) days ago, but was unable to elicit any responses. I am cross-posting it to MO, hoping that it is ...
1
vote
1answer
227 views

Constant group scheme and torsors

Let $X$ be a scheme and $G$ a (commutative) constant group scheme. Consider a $G$-torsor $Y$ for $X$, by which I mean that there is a canonical isomorphism: $$g_Y \colon Y \times_X Y \cong Y ...
1
vote
0answers
187 views

On a property of Riemann Zeta function zeros

Lets consider the function : $$F(x) = \sum_{n=1} (xn)^{-s_0} e^{-nx} $$ with $s_0$ a zero of the Riemann Zeta function in the critical strip. This sum is well defined for $x \in \mathbb{R}^{+*}$. It ...