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
3answers
182 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 ...
3
votes
1answer
142 views

Freiman-isomorphic sets

Haw can we prove that an arbitrary set $A$ of $n$ positive integers is 2-Freiman isomorphic to a subset of {$ 1,2,...,4^{n}$} and $4^{n}$ cannot be improved to $2^{n}$?
5
votes
1answer
293 views

Non-existence of a prime generating polynomial recurrence relation

Let $f\in \mathbb{Q} [x]$ be a polynomial, and $a_0 = a$ be an arbitrary integer. Let us define a sequence $\{a_n \} $ by the recurrence relationship : $$a_n = f(a_{n-1} ). $$ I want to show that $a_n ...
2
votes
1answer
269 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: ...
3
votes
1answer
298 views

What do we know about these subgroups of $S_n$?

For each positive integer $n$, write $S_n$ for the symmetric on $n$-letters. Suppose that $m | n$ is a proper divisor of $n$, and write $n = km$. Consider the element $$\displaystyle u(m,n) = ...
10
votes
1answer
381 views

Galois representations for the curve $y^{2} = x^{3} - x$

Let $E / \mathbb{Q}$ be the elliptic curve given by $y^{2} = x^{3} - x$. I would like to know explicitly what the field of all $2$-power torsion looks like, as well as the image in ...
0
votes
2answers
479 views

What is wrong with this counterexample to the Weak Bunyakovsky's conjecture and reformulation of Bunyakovsky's conjecture?

From HYPOTHESIS H AND AN IMPOSSIBILITY THEOREM OF RAM MURTY. On p. 13 BUNYAKOVSKY’S CONJECTURE ( WEAK FORM ). Let $f$ be a polynomial with integer coefficients and positive leading coefficients ...
33
votes
1answer
1k views

The Bourgain-Demeter-Guth breakthrough and the Riemann zeta function?

Yesterday Bourgain, Demeter and Guth released a preprint proving (up to endpoints) the so-called main conjecture of the Vinogradov's Mean Value Theorem for all degrees. This had previously been only ...
1
vote
0answers
119 views

Gromov-Witten invariants for arithmetic surfaces counting sections passing through points

Suppose we are given an arithmetic surface, $X\to \text{Spec}\mathbb{Z}[1/N]$ smooth and quasi-projective, and a finite set of closed points all in different vertical fibers. Can we count the number ...
-4
votes
2answers
135 views

Diagonal argument for even perfect numbers

Following this, let's define the notion of perfect sequence as follows: $(u_{i})_{i}$ is a perfect sequence if and only if it is the sequence of divisors of an even perfect number in increasing order ...
12
votes
1answer
725 views

On cubic reciprocity for $x^3+y^3+z^3 = 996$?

I. The Diophantine equation, $$x^3+y^3+z^3 = 3w^3\tag1$$ with $x\geq y \geq z$ and $w=1$ has only two known solutions, namely $1,1,1$ and $4,4,-5$. Are there larger ones? As Noam Elkies points out ...
4
votes
2answers
242 views

3-term arithmetic progressions of terms as frequent as primes

Let $\ p_1\ < p_2 < \ldots\ $ be the sequence of all primes $\ (2\ 3\ 5\ \ldots)$. Let $\ x_1 < x_2 < \ldots\ $ be an arbitrary increasing sequence of positive integers such that $\ ...
1
vote
1answer
102 views

construct a Hecke character in MAGMA with given infinity type

I need to do some numerical computation on special values of a Hecke L-function $L(s,\chi)$. To do this, I want to construct a Hecke character in MAGMA, given that I know its infinity type. In other ...
21
votes
2answers
1k views

A conjecture based on Wilson's theorem

Definitions: Lagrange's theorem implies that for each prime $p$, the factors of $(p − 1)!$ can be arranged in unequal pairs, with the exception of $±1$, where the product of each pair $≡ 1 \pmod p$. ...
4
votes
0answers
75 views

Local factors of Tamagawa measure

This is a reference request to some computations which I hope can be found in the literature somewhere. Let $G\subset GL_n$ be a semisimple linear algebraic group over $\mathbb Q$. The Tamagawa ...
6
votes
1answer
232 views

Why are some solutions of these diophantine equations off the usual patterns?

This is inspired by a recent question about complete multipartite integral graphs. I am wondering if more can be said about tripartite integral graphs with block sizes $a<b<c$. It is easy to see ...
8
votes
0answers
174 views

Precise relationship between “finite” Fourier analysis and Galois theory in solving the cubic?

Suppose we want to solve$$x^3 - ax^2 + bx - c = 0.$$We know a priori that this can be factored as $(x - r_0)(x - r_1)(x - r_2)$; by Vieta's formulas, we know$$a = r_0 + r_1 + r_2,\quad b = r_0r_1 + ...
5
votes
0answers
1k views

Asymptotic Robin inequality and RH [closed]

There exists several equivalent formulations of RH. Among them, there is a criterion of Robin that describes a bound on the growth rate of the sum-of-divisors function $\sigma$. Apparently yesterday ...
3
votes
1answer
249 views

Is this problem of Schinzel and Tijdeman misquoted? It appears easy with Pell equations

In Diophantine equations over the twentieth century: a (very) brief overview , p. 5 Problem Let $f(x) \in \mathbf{Z}[x]$ be an irreducible polynomial of degree at least 2. Do the Diophantine ...
0
votes
0answers
75 views

Sum of reciprocals of primitive sequences with distinct prime factors

In a previous mathoverflow question here a construction of a primitive sequence $1<a_1<\cdots<a_k\leq n$ formed by including all the integers in $[1,n]$ with exactly $k$ prime divisors ...
0
votes
1answer
87 views

Fractional equations [closed]

Let’s have the equation $a^k+\frac{k \cdot a^k}{x \cdot y^n}=m^k$ where $k≥2$ and $x≠y$ and $a, k, x, y, n, m$ positive integers greater than zero. If $x \cdot y^n = f \cdot a^g$ where $f$ and $g$ ...
8
votes
2answers
434 views

Fourier transform of the critical line of zeta?

This was asked on MSE and got a lot of upvotes but no answers, so I'm posting it here. Is there a known expression for the (distributional) Fourier transform of the Riemann zeta function, taken along ...
1
vote
1answer
257 views

How can one show that the hyperelliptic curve $y^2 = x^{p} + \frac{1}{4}$ has only one positive rational solution for every prime $p>3$?

Without applying Fermat's Last Theorem, how can one show that the hyperelliptic curve $y^2 = x^{p} + \frac{1}{4}$ has only one positive rational solution $(x,y) = (0, \frac{1}{2})$ for ever prime $p ...
1
vote
1answer
166 views

Polygonal Mersenne numbers [closed]

I posted the same question on Math SE since this one got put on hold. Link to Math SE question:Polygonal Mersenne numbers Polygonal numbers are of the form $\cfrac {n^2(s-2)-n(s-4)}{2}$, where $s$ ...
3
votes
0answers
131 views

Almost primes in short intervals

Define an integer $n$ to be a $k$-almost prime if it has at most $k$ distinct primes factors. A detecting function for the set of such numbers is the generalized von Mangoldt function given by ...
1
vote
1answer
110 views

Upper and lower bounds for $|L(1+it,\chi)|$ for complex primitive character $\chi$?

I would guess this is some standard fact related to the zero-free region. But cannot find it in the textbooks I read.
13
votes
3answers
406 views

Explicit construction of an element of ${\rm GL}(2, p)$ of order $p+1$

It is well-known that the order of $GL(2, p)$ is $(p^2-1)(p^2-p) = (p-1)^2(p+1)p.$ It is easy to construct matrices of orders $(p-1)$ and $p$ (diagonal and parabolic, respectively), but the only way ...
5
votes
0answers
94 views

Property of Dirichlet character

Let $m$ be an integer prime to $p$ such that $\chi^m = \chi_0$ on elements of $\mathbb{F}_p^\times$. We let $\zeta_m$ be a primitive $m$th root of unity. For $b$ any integer prime to $m$ define ...
5
votes
1answer
341 views

The Weil numbers and modulus of an elliptic curve

I have an ignorant question about elliptic curves which I'll be slightly imprecise about. If I have an elliptic curve $X$ defined over $\mathbb Z$, I can base change to $\mathbb C$, and then ...
3
votes
2answers
260 views

Sumsets with distinct numbers, upper bound for maximum element

Let $A$ be a finite set of positive natural numbers with $n$ elements, $|A|=n$, with the property that all sums of two (not necessarily different) elements are distinct, or in the usual notation for ...
12
votes
1answer
1k views

Is Lehmer's polynomial solvable?

The degree 10 polynomial $$\displaystyle x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1$$ given by D.H. Lehmer in 1933 has the property that its largest real root, $\beta = 1.176280 \cdots$ is ...
3
votes
1answer
214 views

Some divisibility constraints in Frobenius coin problem

Let's say that the linear form $ax+by$ represents $n$ if $ax+by=n$ for some positive integer $x$ and $y$. Call a pair $(a,b)\in\Bbb N\times\Bbb N$ with $\mathsf{gcd}(a,b)=1$ excellent if linear form ...
2
votes
2answers
283 views

Cyclotomic polynomials with 7$^{th}$ coefficient greater than 1 in absolute value

It is known that the seventh coefficient of $\Phi_{105}(x)$ is $-2$ and that's the first occurrence of a coefficient with absolute value greater than $1$ for a cyclotomic polynomial. When I did a ...
6
votes
0answers
154 views

Characterizing regular Galois extensions by the set of their specializations

Let $E$ and $F$ be two regular Galois extensions of $\mathbb{Q}(t)$ with group $G$, and let $E_{t_0}$ resp. $F_{t_0}$ be the residue fields corresponding to specializing $t\mapsto t_0\in \mathbb{Q}$. ...
8
votes
1answer
359 views

Dessins d'enfants and absolute Galois group

I would like to know what is the recent progress about the group homomorphism $$ \mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\rightarrow \mathrm{Out}(\hat{F_{2}})$$ ...
3
votes
1answer
164 views

Primitive sequence $a_i$ attaining Pillai's bound on $\sum_{i} 1/a_i$

A primitive sequence $1<a_1<\ldots<a_k\leq n$ is a sequence of integers no one of which divides any other, investigated by Erdos, Behrend and others, over the last 80 years. In fact, $\max ...
1
vote
1answer
49 views

Construction of $n$ makes $s_2(nk)<s_2(n)$

$s_2(n)$ denotes the sum of the standard base-2 digits of $n$. For a fixed odd number $k>1$, can we construct $n\in \mathbb{Z}^+$, to make $s_2(nk)<s_2(n)$? To clarify, that's not $s_2(nk) \lt ...
3
votes
2answers
301 views

Are there other integer solutions to the equation $9x^3 -1 = y^3$ besides $(x,y) =(1,2)$ and $(0, -1)$? [closed]

Does the above Diophantine equation have other integer solutions besides $(x,y)=(1,2)$ and $(x, y) = (0, -1)$?
9
votes
1answer
212 views

Fundamental units with norm $-1$ in real quadratic fields

If we have distinct primes $p \equiv q \equiv 1 \pmod 4,$ with Legendre $(p|q) = (q|p) = -1,$ there is a solution to $u^2 - pq v^2 = -1$ in integers and the fundamental unit of $O_{\mathbb ...
4
votes
1answer
251 views

Problem related to Frobenius coin problem

Let's say that the linear form $ax+by$ represents $n$ if $ax+by=n$ for some positive integer $x$ and $y$. Call a pair $(a,b)\in\Bbb N\times\Bbb N$ with $\mathsf{gcd}(a,b)=1$ good if, for any ...
17
votes
2answers
399 views

“Fractally self-similar” numbers

This is another question about visualization of Ford circles, the previous one being Confusion with practically implementing rational approximations. Here is an output of zooming into Ford circles at ...
1
vote
0answers
74 views

Consecutive integers divisible by consecutive small numbers

Given $n$, what is the largest set of consecutive integers in $[n,2n]$ can we have so that each integer is divisible by a distinct element from $[\log n,2\log n]$ (no partiular order)? So apriori I am ...
2
votes
2answers
345 views

Polynomials which always assume perfect power values

Let $f(x)$ be a non-constant polynomial with integer coefficients. It is a well-known result that if $f(n)$ is a square for all integers $n$, then $f$ must in fact be the square of a polynomial (see, ...
1
vote
1answer
169 views

Least simultaneous quadratic non-residue

Suppose $p,q$ are distinct primes with least quadratic non-residues $n_p$ and $n_q$ respectively. Can one bound the least $n$ for which $\left(\frac{n}{p}\right)=\left(\frac{n}{q}\right)=-1$ in terms ...
5
votes
0answers
64 views

Some questions about the Lévy monoid of certain densities

Let $\bf H$ be a set, $f: \mathcal P({\bf H}) \rightharpoonup \bf R$ a partial function, and $\mathcal{D}$ the domain of $f$. Next, denote by $\mathcal M(f)$ the set of all (total) functions $\theta: ...
6
votes
3answers
1k views

Euler's constant: irrationality and proof theory

Let γ represent Euler's constant. Is there a real number x such that there is a proof within Zermelo-Fraenkel set theory (ZF) that x is irrational and there is also a proof within ZF that γ + x is ...
6
votes
2answers
258 views

Does the limit of this product over primes converge for all $\Re(s) > \frac12$?

Numerical evidence suggests that: $$\displaystyle F(s):= \lim_{N \to \infty}\, \ln^s\left(p_N\right)\, \prod_{n=1}^N \left(\dfrac{\left(p_n-1\right)^s}{p_n^s-1} -\frac{1}{p_n^s}\right)$$ with $p_n$ ...
10
votes
1answer
203 views

Confusion with practically implementing rational approximations

Writing a program visualizing Ford circles I've encountered a seemingly purely programmatic puzzle but then gradually realized there are some mathematical aspects of it which I don't understand. Let ...
0
votes
0answers
48 views

$\mathsf{GCD}$s of random linear form

Given $a,b\in\Bbb N_{<M}$ where $M\in\Bbb N_{>\exp(18)}$ is arbitrary with $(a,b)=1$, the probability that $\mathsf{gcd}(ax_1+by_1,ax_2+by_2)=1$ where $x_1,x_2,y_1,y_2\in\Bbb N_{>\ln M}$ is ...
0
votes
0answers
77 views

Probability distribution associated with total divisors of an integer

Is there a generalization to https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac_theorem which gives distribution function for $$\omega(n)=\big|\{d\in\mathsf{prime}:d|n\}\big|$$ where ...