nt.number-theory's questions - English 1answer

9.979 nt.number-theory questions.

Suppose I have a hypersurface $V(F) = \{ \mathbf{x} \in k^n: F(\mathbf{x}) = 0 \}$, where $F$ is a homogeneous form of degree $d > 1$. I would like to show that there exists some diagonal form $D(...

Let $f_q(x)$ be the generating function of the sequence $(q;q)_n$: $$f_q(x):=\sum_{n=0}^{\infty} (q;q)_n x^n,$$ where $(q;q)_n: = (1-q)(1-q^2) \cdots (1-q^n)$ with convention $(q;q)_0$:=1. Let $g_q(...

Is it true that $$\operatorname{li}(x)-\operatorname{Ri}(x) \sim \frac{1}{2}\operatorname{li}(x^{1/2}) \ (x \to \infty),$$ where $$\operatorname{Ri}(x) = \sum_{n = 1}^\infty \frac{\mu(n)}{n} \...

I am looking for an electronic copy of this volume: Advanced studies in Pure Mathematics, Volume 17 Algebraic Number Theory - in honor of K. Iwasawa Edited by J. Coates, R. Greenberg, B. Mazur and I. ...

The idea for this comes from the twin prime conjecture, where the heuristic evidence seems just so overwhelming, especially in the light of Zhang's famous result from 2014 about Bounded gaps between ...

Let $\pi_p$ be a ramified representation of $GL(n,\mathbb{Q}_p)$. Let $\chi_p$ be a ramified representation of $GL(1,\mathbb{Q}_p)$. Is it generally known that $L(s,\pi_p\times \chi_p)=1$ if $\...

I have always been intrigued by the fact that Riemann's hypothesis is equivalent to the assertion that the inequality $$\frac{\sigma(n)}n<e^\gamma \log\log n \qquad\qquad\qquad(*)$$ holds for all $...

I start think in finding a small divisor $p$ of $M_n$ then I obtain that $p\mid (3^{M_{n}-1}-1)$ hence $ord_{p}(3)\mid M_{n}-1$ Now can I prove p is the same as $M_{n}$ so that $M_{n}$ is not ...

Suppose $V$ is an algebraic variety over a number field $K$. Then, the absolute Galois group $G_K$ of $K$ acts by outer automorphisms on the etale fundamental group $\pi_1(V_{\bar{K}})$ where $\bar{K}$...

In trying to understand the import of Akshay Venkatesh' most recent work I found myself wondering anew about that old gnawing mystery: why Langlands? Why should arithmetic of polynomial equations over ...

Let $n$ be a square-free integer. Then for a given integer $m$, $m$ is a square modulo $n$ if and only if the sum $$\displaystyle \sum_{d | n} \left(\frac{m}{d}\right) > 0.$$ In fact one can ...

Let $f$ be a modular function of level $N>1$ and let $\gamma\in SL_2(\mathbf Z)$. Write $$f(\gamma\tau)=j(\gamma,\tau)f(\tau).$$ First question What can we say in general about the factor $j(\...

let $p,q , r, s,t$ be primes , I w'd like to know if the following equation has solutions in integers : for $n\geq 2: p^n+q^n=r^n+s^n+t^n$ exactly if there are finitely many integers $n$ satisfy ...

Consider the following Diophantine equation $$x^2+x+1=(a^2+a+1)(b^2+b+1)(c^2+c+1).$$ Assume also that $x,a,b,c,a^2+a+1,b^2+b+1, c^2+c+1$ are all primes. We'll call such a quadruplet $(x,a,b,c)$ a ...

I am looking for interesting examples of categorical representations of absolute Galois groups of arithmetic fields. Pointers to the literature would be appreciated.

Robin's theorem (1984) states that $$ \sigma(n) < e^\gamma n \log \log n$$ for all $n > 5040$ if and only if the Riemann hypothesis is true. Recall that $γ$ is the Euler–Mascheroni ...

What is the name for map $f:\mathbb{N}\rightarrow\mathbb{N}$ (from natural numbers into natural numbers) with the following propeties: 1) $f$ is injective 2) $f(n)\geq n$ for every $n$?

I have a polynomial $f(x)=a^2x^2+bx+c\in\mathbb Z[x]$ with $f(x)$ not a constant times a square and $abc\neq0$ and I want to know how many $x$ between $-a$ and $a$ the polynomial is a perfect square. ...

The question is: for any finite group, $G$, and any finite set of primes (of $\mathbb{Z}$), $P$, is there a number field $K$, such that there is a regular $G$-Galois extension of $\mathbb{P}^1_K$, and ...

Take two distinct coprime integers $a,b$ and take $q=a^2+b^2$. Consider the equation $(xy^{-1})^2\equiv-1\bmod q$. The solutions are two roots which correspond to $(x,y)=(a,b)$ and $(x,y)=(b,a)$. Look ...

I've often seen it stated (in vague terms) that there's a Fourier duality between the set of prime numbers and the set of nontrivial Riemann zeta zeros. Because there are various explicit formulae ...

Suppose $G$ is a connected reductive group over an algebraically closed field. Then given a maximal torus $T$, we can define a Weyl group $W$ and consider $T^W$, the Weyl-invariants of $T$. This ...

In number theory there is often an analogue between statements which holds over a number field (that is, a finite field extension $K/\mathbb{Q}$) and function fields (that is, finite extensions of the ...

An Egyptian fraction expansion is a sum of reciprocals of integers, for example: $$\frac{4}{17} = \frac{1}{5} + \frac{1}{29} + \frac{1}{1233} + \frac{1}{3039345}$$ Every positive rational number $p/...

[Please allow me a note: The way desribed below allows to depict functions $f:X^2 \rightarrow Y$ completely in two dimensions (without hiding or omitting any information). This allows for depicting ...

Let $\Gamma$ be the modular group and $\mathcal M_n$ the set of all primitive matrices with determinant $n\geq1$. Recall that a primitive matrix has relatively prime entries. The modular group $\...

I have read in https://www.sciencealert.com/mathematicians-discover-a-strange-pattern-hiding-in-prime-numbers that says: "But this doesn't explain the magnitude of the bias the team found, or why ...

In his 2004 paper Conformal Field Theory and Torsion Elements of the Bloch Group, Nahm explains a physical argument due to Kadem, Klassen, McCoy, and Melzer for the following remarkable identity. Let $...

Let $d(n)$ denote the number of positive divisors of the positive integer $n$. Pick some positive $X,h \in \mathbb{R}$ and consider the sum $$ S(X,h) := \sum_{X \leq n \leq X+ h} d(n).$$ In view of ...

I have a couple of elementary questions regarding automorphy of Galois representations arising from geometric families. Suppose we have an algebraic family of varieties over a number field, and ...

No nontrivial integer solutions to $$ a^5+b^5=c^5+d^5 \qquad (1)$$ are known. (1) has infinitely many solutions in an extension of $\mathbb{Z}$ (root of $9-15x+37x^2 $ ) resulting from a genus 0 ...

Let $p^{(s)}(n)$ be the number of ways of writing the positive integer $n$ as a sum of perfect $s$-powers, where the order does not matter. For example, $p^{(2)}(9) = 4$ since $$9 = 1^2 + 1^2 + 1^2 + ...

If for a given π(prime(n)) or f(π(prime(n))), we can find prime(n), is this enough to help on solve RH?

Along the lines of the general question "How much does the discriminant of a number field reveal about the field?", I was wondering how often it happens that the discriminant of some number field ...

The semiprime $87 = 3*29$ has a curious property: it's the fact that both $87^2 + 29^2 + 3^2 = 8419$ and $87^2 - 29^2 - 3^2 = 6719$ are prime numbers. This intrigued me and led me to wonder if ...

Is there a number that can be represented as a sum of two fourth powers in three different ways?

I hope it is OK to ask the following reference request. If my question is not suitable, please let me know and I will do my best to modify it! Let $N\in\mathbb{N}$, let $q$ be a point in the open ...

According to wiki: https://en.wikipedia.org/wiki/Dedekind_eta_function, Dedekind eta function is defined in many equivalent forms. But none of them is an explicit description (say in algorithmic ...

i.e. does there exist an integer $C > 0$ such that $11, 11 + C, ..., 11 + 10C$ are all prime?

Can you provide a proof or a counterexample for the following claim : Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$ Let $N=k\cdot 2^n+1$...

Given integers $a,\ell$ and prime $p$ we need to find the roots of the algebraic equation $x^\ell\equiv a\bmod p$. We know there are at most $\ell$ such $x$. What is the best method to find all such ...

This is a follow-up to an earlier question. The answer to that question was found on this page. The discussion on OEIS seems to suggest that, for any prime $p$, there should exist a $p$-length ...

John Derbyshire in his book PRIME OBSESSION says on page 366: CHAPTER 3 10. "Here is an example of e turning up unexpectedly. Select a random number between 0 and 1. Now select another and add ...

This is a very basic question of course, and exposes my serious ignorance of analytic number theory, but what I am looking for is a good intuitive explanation rather than a formal proof (though a ...

Let $A$ be a set of integers. In our recent researches, we've faced to the following property and definition: We say $A$ has infinite difference length, if (a) For every integer $n$ there exist a ...

John Derbyshire in his book PRIME OBSESSION says on page 366: CHAPTER 3 10. "Here is an example of e turning up unexpectedly. Select a random number between 0 and 1. Now select another and add ...

Let $(a,b)_K$ be the quadratic Hilbert symbol in a local field $K$. Let $a$ be a rational number. By a consequence of the quadratic reciprocity law we have: $$\prod_{p} (a,-1)_{\mathbb{Q}_p}=\mathrm{...

Let $K$ be a local field with residue field $k$ and $E/K$ an elliptic curve. I'm interested for which $K$ and $E$ the group of torsion points on the curve is finite. I can prove that this group is ...

Let $\pi(x)$ denote the number of primes $\leq x$. What is the asymptotic form for $$\sum_{r=1}^{\pi(x)-1} \Bigg(\frac{(\pi(x))!}{r!(\pi(x)-r)!}-\frac{(\pi(x))!}{(r-1)!(\pi(x)-r+1)!} \Bigg) $$ ? The ...

The setup is as follows: $k/\mathbb{Q}_p$ is a finite extension, $\mathfrak{p}$ is the maximal ideal of $\mathcal{O}_k$, $q=\#(\mathcal{O}_k/\mathfrak{p})$ $k'/k$ is a finite unramified extension of ...

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