Questions tagged [nt.number-theory]

Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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29 views

Good software for shortest vector for non-full rank lattices?

what is a good algorithm for shortest vector in non-full rank lattices in say dimensions $8$ to $64$? Is there a software package available?
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45 views

Reference describing supersingular elliptic curves over algebraically closed field in characteristic 2

I'm looking for a reference for the fact that over an algebraically closed field of characteristic two, there is (essentially) only one supersingular elliptic curve. This fact appears on Wikipedia, ...
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56 views

Ekedahl sieve for composite moduli

The following version of a theorem of Ekedahl, known as the Ekedahl sieve, can be found in this paper by Bhargava and Shankar for example, is stated as follows: let $B$ be a compact subset of $\mathbb{...
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1answer
135 views

Irreducibility of a polynomial when the sum of its coefficients is prime

I came up with the following proposition, but don't know how to prove it. I used Maple to see that it holds when $ a + b + c + d <300 $. Let $a,b,c$ and $d$ be non-negative integers such that $d\...
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36 views

Extension of closed-form solvability of polynomial equations of x and exp to rational expressions?

Is my conjecture below true? It's a new conjecture. It's an extension of Lin's theorem in [Lin 1983] which is cited in [Chow 1999] - see both references at the bottom of this page. It seems that Ferng-...
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54 views

What are the asymptotics for $\sum_{r=0}^{x-1} \sum_{|\rho|\leq x} \frac{(x+r)^{\rho+1}}{\rho}$?

Let $f(x)=\sum_{r=0}^{x-1} \sum_{|\rho|\leq x} \frac{(x+r)^{\rho+1}}{\rho}$, where $\rho$ denotes a non-trivial zero of the Riemann zeta function. What are the upper and lower bounds for $f(x)$ ? ...
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2answers
269 views

Is the number of solutions of $\phi(x)=n!$ bounded? If yes, what is its bound?

Pillai showed in 1929 that the function $A(n)$ giving the number solutions of the equation $\phi(x)=n$ is unbounded in (S. Pillai, On some functions connected with $\varphi(n)$, Bull. Amer. Math. Soc. ...
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85 views

Is new n-conjecture as follows correct?

Given a positive integer $P>1$, let its prime factorization be written $$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$$ Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,a_k)$ ...
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37 views

Radical Extensions - gcd and lcm [closed]

Given two natural numbers $m,n$ and say $\gcd(m,n)=g$, I would like to show that $$ \Bbb Q(\xi _n )\cap \Bbb Q(\xi _m ) = \Bbb Q(\xi _g ). $$ One direction is trivial, and to show the other one I ...
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1answer
88 views

Is there any relation between $h_{d_1},h_{d_2}$ and $h_{d_1d_2}$?

$h_{d_1}, h_{d_2}$ and $h_{d_1d_2}$ are class number of $\Bbb Q(\sqrt{d_1}),\Bbb Q(\sqrt{d_2}),\ and \ \Bbb Q(\sqrt{d_1d_2})$ respectively.
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77 views

In how many ways to generate a vector with odd/even entries

Let $\bar v=(k_1, \ldots k_n)$ be a vector with entries $k_i$ from $\{0, \ldots, K\}$ with $K \in N$ and such that $\sum_{i=1}^n k_i=K.$ Find in how many ways we can generate vector $\bar v$ so that ...
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85 views

Is this number theoretic possibility true? [closed]

$a,b,T,k$ are fixed and only $u,v,t,M$ changes and $u,v$ are the only independent or free variables. Let $au+bv+T=L$ with $a,b>u,v>0$, $(au+bv)/2<T<2(au+bv)$ and $a,b$ coprime and in $a,b\...
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1answer
78 views

Sieving by composite moduli

A traditional sieve gives a bound on the number of integers $n$ in an interval (say $I=[0,N]$) such that $$n\not\in S_p \mod p$$ for every prime $p$ in a set $\mathcal{P}$, where $S_p\subset \mathbb{Z}...
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0answers
44 views

Probability of factor of particular size

What is the probability that an integer $a$ picked uniformly in $[t/2,t]$ has a factor in $[t^{\alpha},2t^{\alpha}]$ where $\alpha\in(0,1)$? I am interested when $\alpha=1/3$ but if there is a general ...
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0answers
73 views

Asymptotic for the probability that a number has $k$ prime factors less than $Q$

If we let $\omega_Q(n)$ denote the number of distinct prime factors of $n$ less than a bound $Q$, then what asymptotic formulas exist for $\Pr_{n\in\mathbb{N}}[\omega_Q(n)=k]$ as $Q\to\infty$ if $k$ ...
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0answers
113 views

How smooth can this be?

If $a$ is an even integer then how smooth can $a^2-1$ be? Approximately how many integers in $a\in[0,t]$ are there such that $a^2-1$ is $k$-smooth?
2
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1answer
178 views

Curious inversion formula in additive combinatorics

Let $S$ be an infinite set of positive integers, and $T=S+S=\{x+y, \mbox{ with } x,y\in S\}$.We definte the following functions: $N_S(z)$ is asymptotic continuous version of the function counting the ...
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0answers
51 views

Divisibility between values of polynomials

In this paper Corvaja and Zannier considered the following problem. Let $K$ be a number field, and let $S$ be a finite set of places of $K$ containing the infinite places and let $\mathcal{O}_S$ be ...
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92 views

Primes which do not divide certain homogeneous polynomials

It is known that if $x^2 + y^2 = z^2$ is a primitive Pythagorean triplet then $z$ is not divisible by any prime of the form $4k-1$. The following is a generalization of this classical result which ...
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2answers
335 views
+100

Origin and variations of problem on $4xy-x-y$ being square

One of the forms in which the Diophantine equation in question can be found in the literature is this: Solve the equation \begin{eqnarray}z^{2} = 4xy-x-y \qquad \qquad (\ast)\end{eqnarray} in ...
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1answer
107 views

Density of integral points on affine cubic surfaces of a certain type

Let $Q(x,y,z)$ be a cubic polynomial with integer coefficients, such that the terms $x^3, y^3, z^3$ do not appear. That is, it is at most quadratic in each of the variables $x,y,z$. Is there a general ...
2
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1answer
95 views

Hecke eigenform with integer Fourier coefficients

Is it true that for any even $k$ and $N,$ there always exist a Hecke eigenform with integer Fourier co-efficient of weight $k$ and level $N$ ?
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79 views

Is there a conjectured uniform Lindelof hypothesis for Hurwitz zeta functions

Consider $\zeta(s, a) = \sum_{n=1}^{\infty} (n+a)^{-s}$ (alternatively, consider its functional equation Dirichlet series $\sum_{n=1}e(a n) n^{-s}$). What is the expected growth-rate of $\zeta(1/2 + ...
1
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1answer
429 views

Paradox in additive combinatorics

Let $S$ be an infinite set of positive integers. Let us define the following quantities: $N_S(z)$ is the number of elements of $S$, less or equal to $z$ $r_S(z)$ if the number of positive integer ...
9
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1answer
248 views

Cusp forms with integer Fourier-coefficients

Let $S_k(\Gamma_1(N))(\mathbb{Z})$ be the set of modular forms of weight $k$ and level $N$ with integer Fourier coefficients. Then is true that any cusp form can be written as $\mathbb{Q}$ linear ...
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117 views

On the Dirichlet divisor problem. Proof that $\Delta(n) = O(n^{\frac{1}{4} + \epsilon})$?

Hello dear mathematicians! I have a few questions regarding my current work (paper) on counting the number of lattice points under a hyperbola $\frac{n}{xy},\; 1 \leq x \leq n,\; 1 \leq y \leq n$. ...
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0answers
87 views

Freys elliptic curves and Hilbert spaces?

Consider the Frey-Hellegouarch curve given $a,b$ positive rational numbers: $$y^2= x\left(x-\frac{a}{\gcd(a,b)}\right)\left(x+\frac{b}{\gcd(a,b)}\right)$$ The j-invariant is given by: $$j(a,b) = \frac{...
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51 views

Is the cohomology of Hilbert modular surfaces spanned by special cycles?

We consider the Hilbert modular surface $X$ that parametrize abelian surface with real multiplication by $\mathcal{O}_F$ and $\mathfrak{a}$-polarization, where $F$ is a real quadratic field with ...
7
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1answer
557 views

Is there a collection of evidence and heuristic arguments against the Riemann hypothesis? [closed]

There is undoubtedly an overwhelming collection of evidence for the Riemann hypothesis. However, is there any evidence against it ?
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78 views

On the function $f(\sigma)=\int_{-\infty}^{\infty} | \frac{1}{(\sigma + it)\zeta(\sigma + it)}|^{2} \mathrm{d}t$

Define $$f(\sigma)=\int_{-\infty}^{\infty} \Big| \frac{1}{(\sigma + it)\zeta(\sigma + it)} \Big|^{2} \mathrm{d}t$$ where $\zeta$ denotes the Riemann zeta function and $i$ the imaginary unit. Is $f(\...
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0answers
66 views

Is there any good reference on the Bayesian view that can be helpful for reading papers on the number theory using heuristic arguments?

Nowadays there are many papers on the number theory using heuristics. I have read some of them. But I have no clear understanding of the Bayesian Probability(subjective probability). The concept of ...
3
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1answer
128 views

Mean square estimate for the Kloosterman sums

For $m,n\in \mathbb{N}$, denote the Kloosterman sum $$S(m,n;c)=\sum_{a\bmod c}e\left( \frac{ma+n \overline{a}}{c}\right),$$where $\overline{a}$ denotes the multiplicative inverse of $a\bmod c$. Does ...
1
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1answer
142 views
+50

Integer partitions into restricted parts

Given a linear diophantine equation $$x_1+\dots+x_n=m\leq nn'$$ how many solutions does it have with each $x_i\in[0,n']\cap\mathbb Z$? Looking for asymptotics that parametrizes well with both $n$ and $...
7
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0answers
180 views
+100

Full measure properties for Zariski open subsets in $p$-adic situation

Let $F$ be a $p$-adic field and let $X$ be a smooth integral variety over $F$ (I am chiefly interested in the case when $X$ is a connected reductive group over $F$). Let $U$ be a non-empty open subset ...
4
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2answers
197 views

A generalization of strong primes

In this post we denote the sequence of prime numbers as $p_k$ for integers $k\geq 1$. I don't know if the following definition is in the literature. Definition. We define the $\theta$-strong primes, ...
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97 views

General asymptotic result in additive combinatorics (sums of sets)

Let $S_1,\cdots,S_k$ be $k$ infinite sets of positive integers. Let $N_i(z)$ be the numbers of elements in $S_i$ that are less or equal to $z$. Let us further assume that $$N_i(S) \sim \frac{a_i z^{...
5
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1answer
144 views

On the density map of the abundancy index

Let $σ$ be the sum-of-divisors function. Let $σ(n)/n$ be the abundancy index of $n$. Consider the density map $$f(x) = \lim_{N \to \infty} f_N(x) \ \ \text{ with } \ \ f_N(x) = \frac{1}{N} \#\{ 1 \...
6
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1answer
232 views

When is the characteristic polynomial of the character table of a cyclic group irreducible?

Let $G=C_n$ be the cyclic group and $f_n$ the characteristic polynomial of its character table (over $\mathbb{C}$) in the ordering so that the character table is given by the discrete Fourier ...
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1answer
139 views

Polynomials of minimum degree that interpolate primes in intervals

Given an interval $[a,b]$ what is the minimum degree of univariate polynomials in $\mathbb Q[x]$ that passes through all primes between $a$ and $b$ (denoted by $\mathbb P[a,b]$ with total number of ...
7
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2answers
240 views

Q-curves and twisting

An elliptic curve $E$ over $\overline{\mathbb{Q}}$ is called a $\mathbb{Q}$-curve if it is isogenous (over $\overline{\mathbb{Q}}$) to all its Galois conjugates -- see Are Q-curves now known to be ...
3
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1answer
155 views

P-adic distance between solutions to S-unit equation

Let $p$ be a fixed prime number and $S$ is a finite set of prime numbers which does not contain $p$. A theorem of Siegel asserts that the number of solutions to the $S$-unit equations are finite; that ...
7
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2answers
322 views

Abundancy index and non-solvable finite groups

Let $\sigma$ be the sum-of-divisors function. A number $n$ is called abundant if $\sigma(n)>2n$. Note that the natural density of the abundant numbers is about $25 \%$. The abundancy index of $n$ ...
3
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2answers
152 views

Simultaneous similarity of matrices over finite fields

Suppose $A,B\in SL(3,F_q)$, where $F_q$ is the finite field of order $q$ and $SL(3,F_q)$, the group of matrices with determinant one and entries from $F_q$ , are such that $A$ has eigenvalues in $F_q$...
7
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2answers
362 views

A variant on Wieferich primes

Recall that a Wieferich prime is a prime number $p$ such that $2^{p-1} \equiv 1 \bmod p^2.$ It is not known whether there are infinitely many Wieferich primes, nor whether there are infinitely many ...
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0answers
62 views

$t$-balanced numbers

Disclaimer: throughout this question, we'll assume the truth of Goldbach's conjecture. For $n$ a large enough composite positive integer, write $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$, $...
2
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1answer
168 views

Pell equation and quadratic residues

We say an integer $k$ is Pell if there exist some integers $p,q$ such that $$ p^2k-q^2=1 $$ In studying a physics system we ended up with two weaker notions of Pell: We say an integer $k$ is pre-Pell ...
3
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0answers
89 views

What is the multiplicative order of this number

Let $q, r \in \mathbb{P}$ and $r$ is the next prime to $q$. What is the multiplicative order of $r$ modulo $\displaystyle\bigg( \prod_{\substack{p \leq q \\\text{p prime}}} p \bigg)$ ? In other word ...
6
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0answers
146 views

When did the main conjecture in Vinogradov's mean value theorem first appear in literature?

Recently I was asked about the history of Vinogradov's mean value theorem that I was hoping someone here could clarify. Let me first start with some terminology. Let $J_{s, k}(X)$ be the number of $2s$...
12
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2answers
331 views

On the sum of the subgroup orders of a finite group

Let $G$ be a finite group. Consider the function providing the sum of the subgroups orders $$\sigma(G) = \sum_{H \le G} |H|.$$ Note that if $C_n$ is cyclic of order $n$ then $\sigma(C_n) = \sigma(n)$, ...
1
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0answers
175 views

On the error bound for the Prime Number Theorem for arithmetic progressions

Let $\chi$ be a Dirichlet character, $L(s,\chi)$ be the corresponding L-functions and $\Theta_{\chi}$ be the supremum of the real parts of the zeros of $L(s, \chi)$. Define $\pi(x; a, q)$ to be the ...

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