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
11,964
questions
0
votes
0answers
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?
0
votes
0answers
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, ...
0
votes
0answers
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{...
5
votes
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\...
1
vote
0answers
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-...
0
votes
0answers
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 $f(x)$ ?
...
2
votes
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. ...
0
votes
0answers
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 $h(P)=\min(a_1, a_2,\ldots,a_k)$
...
0
votes
0answers
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 ...
1
vote
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.
-1
votes
0answers
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 ...
-4
votes
0answers
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\...
3
votes
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}...
1
vote
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 ...
1
vote
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 $k$ ...
0
votes
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
votes
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 ...
1
vote
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 ...
1
vote
0answers
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 ...
3
votes
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 ...
1
vote
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
votes
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 $N$ ?
5
votes
0answers
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
vote
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
votes
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 ...
-2
votes
0answers
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 $\frac{n}{xy},\; 1 \leq x \leq n,\; 1 \leq y \leq n$. ...
4
votes
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:
$$j(a,b) = \frac{...
5
votes
0answers
51 views
Is the cohomology of Hilbert modular surfaces spanned by special cycles?
We consider the Hilbert modular surface $X$ that parametrize abelian surface with real multiplication by $\mathcal{O}_F$ and $\mathfrak{a}$-polarization, where $F$ is a real quadratic field with ...
7
votes
1answer
557 views
Is there a collection of evidence and heuristic arguments against the Riemann hypothesis? [closed]
There is undoubtedly an overwhelming collection of evidence for the Riemann hypothesis. However, is there any evidence against it ?
-4
votes
0answers
78 views
On the function $f(\sigma)=\int_{-\infty}^{\infty} | \frac{1}{(\sigma + it)\zeta(\sigma + it)}|^{2} \mathrm{d}t$
Define $$f(\sigma)=\int_{-\infty}^{\infty} \Big| \frac{1}{(\sigma + it)\zeta(\sigma + it)} \Big|^{2} \mathrm{d}t$$ where $\zeta$ denotes the Riemann zeta function and $i$ the imaginary unit. Is $f(\...
2
votes
0answers
66 views
Is there any good reference on the Bayesian view that can be helpful for reading papers on the number theory using heuristic arguments?
Nowadays there are many papers on the number theory using heuristics.
I have read some of them.
But I have no clear understanding of the Bayesian Probability(subjective probability).
The concept of ...
3
votes
1answer
128 views
Mean square estimate for the Kloosterman sums
For $m,n\in \mathbb{N}$, denote the Kloosterman sum
$$S(m,n;c)=\sum_{a\bmod c}e\left( \frac{ma+n \overline{a}}{c}\right),$$where $\overline{a}$
denotes the multiplicative inverse of $a\bmod c$.
Does ...
1
vote
1answer
142 views
+50
Integer partitions into restricted parts
Given a linear diophantine equation $$x_1+\dots+x_n=m\leq nn'$$ how many solutions does it have with each $x_i\in[0,n']\cap\mathbb Z$? Looking for asymptotics that parametrizes well with both $n$ and $...
7
votes
0answers
180 views
+100
Full measure properties for Zariski open subsets in $p$-adic situation
Let $F$ be a $p$-adic field and let $X$ be a smooth integral variety over $F$ (I am chiefly interested in the case when $X$ is a connected reductive group over $F$). Let $U$ be a non-empty open subset ...
4
votes
2answers
197 views
A generalization of strong primes
In this post we denote the sequence of prime numbers as $p_k$ for integers $k\geq 1$. I don't know if the following definition is in the literature.
Definition. We define the $\theta$-strong primes, ...
0
votes
0answers
97 views
General asymptotic result in additive combinatorics (sums of sets)
Let $S_1,\cdots,S_k$ be $k$ infinite sets of positive integers. Let $N_i(z)$ be the numbers of elements in $S_i$ that are less or equal to $z$. Let us further assume that
$$N_i(S) \sim \frac{a_i z^{...
5
votes
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
votes
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 ...
-2
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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 ...
1
vote
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
votes
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
$$
In studying a physics system we ended up with two weaker notions of Pell:
We say an integer $k$ is pre-Pell ...
3
votes
0answers
89 views
What is the multiplicative order of this number
Let $q, r \in \mathbb{P}$ and $r$ is the next prime to $q$.
What is the multiplicative order of $r$ modulo $\displaystyle\bigg( \prod_{\substack{p \leq q \\\text{p prime}}} p \bigg)$ ?
In other word ...
6
votes
0answers
146 views
When did the main conjecture in Vinogradov's mean value theorem first appear in literature?
Recently I was asked about the history of Vinogradov's mean value theorem that I was hoping someone here could clarify. Let me first start with some terminology. Let $J_{s, k}(X)$ be the number of $2s$...
12
votes
2answers
331 views
On the sum of the subgroup orders of a finite group
Let $G$ be a finite group. Consider the function providing the sum of the subgroups orders
$$\sigma(G) = \sum_{H \le G} |H|.$$
Note that if $C_n$ is cyclic of order $n$ then $\sigma(C_n) = \sigma(n)$, ...
1
vote
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 ...
