All Questions
69 questions
1
vote
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186
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Existence of Finite Amicable Groups
I'm interested in exploring the concept of "amicable groups" as follows:
Definition. Two finite groups $G$ and $H$ are called amicable groups if:
$G$ is the direct sum of proper subgroups ...
2
votes
1
answer
110
views
Asymptotic behavior in a modular color-cycling problem
Consider the following problem: We have $k$ rooms, each equipped with a light that cycles through three colors – red, green, and blue – in a cyclic order. Initially, all lights are set to red. Each of ...
3
votes
1
answer
655
views
Separating Gamma in two independent functions
I've encountered a problem in my PhD. I would greatly appreciate any suggestions, tips, or comments you might have. The problem is
Let $\Gamma(s,x)$ be the incomplete gamma function. Given integers $n ...
- 41
1
vote
0
answers
59
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A question on generalized bases
I just came to know that it is possible to define a generalized base as an infinite sequence of natural numbers $\mathbf b=(b_1,b_2,\dots)$ where $b_i\ge 2$ for all $i$. With this definition, any $m\...
3
votes
1
answer
855
views
Infinite dimensional lattice for integers and the Riemann hypothesis?
It is known that for each finite set of primes $p$ we have: $\log(p)$ are linear independent over the rational numbers.
We have $\log(ab) = \log(a)+\log(b)$ and $\log(n) = \sum_{p |n}v_p(n) \log(p)$.
...
- 3,064
1
vote
1
answer
472
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Conjectured upper bound on the maximum value of the absolute value of the Möbius function in the poset of multiplicative partitions under refinement
PRELIMINARIES:
Consider the poset $(\mathcal{P}_n, \leq_r)$ of the (unordered) multiplicative partitions of $n$ partially ordered under refinement (for all $\lambda, \lambda’ \in \mathcal{P}_n$, we ...
- 405
3
votes
2
answers
641
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Computing hypergeometric function at 1
I'm looking to compute
$${}_ 3F_ 2\biggl(\begin{matrix} -m-1/2,\ -m,\ k-m+1/2 \cr
1/2-m,\ k-m+3/2\end{matrix};1\biggr)$$
for $m,k > 0$ are positive integers and $0 < k < m$. I'm wondering if ...
- 337
16
votes
2
answers
1k
views
The Stable Set Conjecture
A set $\mathcal S$ of positive integers is called stable if for every fixed positive integer $d$, the relation
$$n\in \mathcal S \iff dn\in \mathcal S$$
holds for almost all positive integers $n$. ...
- 791
2
votes
2
answers
283
views
Expressions for binomial residue sum $\sum_{k=0}^n {n \choose k} x^k \left( \frac{k}{q} \right)$
I'm interested in the sum:
$$\sum_{k=0}^n {n \choose k} x^k \left( \frac{k}{q} \right)$$
where $q$ is a prime number. This is just the binomial expansion with an extra weight on quadratic residues ...
- 591
4
votes
1
answer
260
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Kummer's congruence at $p=3$
Let $B_{2k}$ be the Bernoulli numbers of even index and $\varphi(n)$ be Euler's totient function. We recall one instance of Kummer's congruences: for each integer $m\geq1$ and a prime number $p\geq5$, ...
- 43.2k
3
votes
1
answer
312
views
Congruences for power-sum of divisors
If $\sigma_k(n)=\sum_{d\vert n} d^k$, denote
$$F_1(q)=\sum_{n\geq1}\sigma_1(n)\,q^n \qquad \text{and} \qquad
F_3(q)=\sum_{n\geq1}n\cdot\sigma_2(n)\,q^n.$$
QUESTION. Assume the prime $p$ is either $2,...
- 43.2k
2
votes
0
answers
286
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Is Sturm's theorem able to do these?
$\newcommand{\Ord}{\operatorname{Ord}}$Let $p$ be a positive integer and $F(q)=\sum A(m)q^m$ be a formal power series with integer coefficients. Then $\Ord_p(F(q))$ is defined by
$$\Ord_p(F(q)):=\min\{...
- 43.2k
1
vote
1
answer
115
views
Cardinality of $\{ n_i + i^k: i \in \mathbb{N} \} \cap [1,T]$ where $\{n_i \}$ is all natural numbers in some order
Let $n_1, n_2, ...$ be a sequence of natural numbers such that $\{n_i: i \in \mathbb{N}\}$ as a set is all of natural numbers. Let $k$ be a positive integer. Is is possible to obtain a lower bound of ...
- 3,625
1
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0
answers
123
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On probability of coprimality of a list of numbers
We know $r$ randomly chosen integers are coprime with probability $\frac1{\zeta(r)}$.
Pick a bound $N$ and pick $k\lceil N^{\alpha}\rceil$ uniformly random integers in $[0,N^{\alpha+\beta}]$ where $\...
- 13.9k
1
vote
0
answers
134
views
Number of solutions to a diophantine equation
Given a positive integer $n$, consider the diophantine equation $4x^2+y^2+4x+y=2n$ with solutions in non-negative integers $x$ and $y$.
Define the proportion
$$\delta_n=\frac{\#\{(x,y)\in\mathbb{Z}^2_{...
- 43.2k
4
votes
0
answers
231
views
How big are small inverse powers of 2 mod powers of 3?
Let $a \bmod b$ take value as an integer in $[0, b)$. For any $T \ge 1$, for what $R \in [0, 3^n)$ is
$$\min \{2^{-t}\bmod 3^n: t =1, \dotsc, T\} := \min A_T > R?$$
When $T$ is fixed as $n$ ...
- 459
2
votes
1
answer
215
views
The growth of the number of Fano complete intersection families
I recently calculated the number (possible multidegrees) of Fano complete intersections of dimension $n$ , because I wanted to make the remark that it grows "very rapidly" as $n \rightarrow \...
- 6,995
3
votes
1
answer
372
views
How many ways can $N$ be written as a sum of terms in the form $2^i3^j$?
Given a positive integer $N$, let $f(N)$ be the number of ways $N$ can be decomposed as a sum of terms of the form $2^i3^j$, where each such term appears at most once in the sum. For example, $f(10) = ...
- 1,703
2
votes
1
answer
273
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Primes in modular arithmetic progression
Fix a prime $p$.
I want to get $k<p$ primes $p_1<\dots<p_k$ such that at every $i\in\{1,\dots,k\}$ we have
$$p_i\equiv (2i+1+c)\bmod p$$ where $c$ is fixed and $2k+1+c<p$ holds.
For a ...
- 13.9k
3
votes
0
answers
157
views
Combinatorial interpretation for Möbius-poly-Bernoulli numbers
The Möbius-Bernoulli numbers ,are related to Dedekind Sums
$$\sum_{d|n}\frac{t\mu(d)}{e^{td}-1}=\sum_{k=0}^\infty M_k(n)\frac{t^k}{k!}$$ where $|t|<\frac{2\pi}{n}$, and $M_k(1)=B_k$.
We define the ...
user160903
1
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0
answers
126
views
What is the order of binary partition function when number of partitions is fixed?
Given a number $N$, the number of ways to write it as a sum of powers of $2$ is called the binary partition function of $N$ and is well studied. But if the number partitions are fixed, then how do you ...
1
vote
0
answers
80
views
Number of integers in relation to Pythagorean triples with some quadratic relations
Given an integer $m>0$ possibly composite we can find non-negative integers which are not equivalent to $0\bmod m$ with
$$ab+cd\equiv0\bmod m$$.
Is there any integer quadruples bounded in $[0,m-1]^...
- 1,826
1
vote
1
answer
394
views
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 $...
- 1,826
0
votes
1
answer
260
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Generalized Erdős multiplication table problem
Consider multiplication operation $$f(x_1,\dots, x_k)=\prod_{i=1}^kx_i$$ where $x_i\in\{1,\dots, n_i\}$ with $n_1,\dots, n_k\in\{1,\dots,\infty\}$.
What is the cardinality of the range?
At $k =2$ ...
- 1,826
2
votes
0
answers
156
views
Questions about a certain sequence of naturals generated by primorials
I'm working on the following sequence of naturals (which is NOT listed in OEIS)
$$3,5,11,17,23,29,59,89,119,149,179,209,419,629,839,1049,1259,1469,1679,...$$
whose elements are generated this way
$$3=(...
- 1,008
4
votes
0
answers
191
views
Overview of Combinatorial Technique of "Selberg’s Symmetry Formula"
In the paper entitled "A Computational History of Prime Numbers and Riemann Zeros" (on page 4, click here) it is written about "Selberg’s symmetry formula" that-
Until 1950 it was widely believed (...
0
votes
0
answers
128
views
Number of primes skipped by binomial coefficients?
Take $$B(l,n)=\binom{n+l}{n}$$ and $\mathcal P(t)=\{p\mbox{ prime}:p|t\}$.
What is the cardinality of $\mathcal P(B(l,n))$?
What is minimum cardinality of $L\subseteq\{1,\dots,n\}$ such that $$\...
- 13.9k
12
votes
2
answers
743
views
Smallest set such that all arithmetic progression will always contain at least a number in a set
Let $S= \left\{ 1,2,3,...,100 \right\}$ be a set of positive integers from $1$ to $100$. Let $P$ be a subset of $S$ such that any arithmetic progression of length 10 consisting of numbers in $S$ will ...
- 179
3
votes
4
answers
280
views
Approximately satisfying simultaneous vector linear diophantine equations?
Pick three $a,b,c$ vectors in $\mathbb Z^n$ uniformly with $\max(\|a\|_\infty,\|b\|_\infty)<T$ and $\|c\|_\infty<T^2$ and an $\epsilon>0$.
Assume $a$ and $b$ are coordinatewise coprime (...
- 13.9k
8
votes
1
answer
575
views
Unstable Integers
There is a question that has been bothering my mind for quite a while now. I will present it and my current thoughts and progress on it.
Let the prime factorization of an integer $n$ be
$$n = p_1^{...
- 311
1
vote
0
answers
126
views
How many solutions to $p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n$?
Consider a system of $n$ divisibility conditions on $n$ prime variables:
$$p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n,\;\;\;\;\;1\leq i\leq n,$$
where $a_{i,j}$ are bounded integers. How many solutions ...
- 545
3
votes
0
answers
148
views
Maximum number of integral roots in degree $d$ polynomial?
Given $f(x_1,\dots,x_n)\in\mathbb Z[x_1,\dots,x_n]$ such that
Each coefficient is bound in absolute value by $B$
Degree of each variable in any monomial is bound by $d$
Total degree is $d'$
$f(x_1,\...
- 13.9k
-3
votes
1
answer
269
views
Negative Dirichlet Pigeonhole Principle [closed]
From Dirichlet Pigeonhole Principle if $p$ is a prime and if $a,b\in\mathbb Z$ are in $(0,p/2)$ then there is a $t\in(0,p)\cap\mathbb Z$ such that $\|(x,y)\|_\infty<\lceil\sqrt p\rceil$ holds where ...
- 13.9k
0
votes
1
answer
298
views
Discrepancy in non-homogeneous arithmetic progressions
I have a doubt, Roth's discrepancy theorem [1] says that there is a subset of arithmetic progressions $A \in [n]$ where any function $f:N\rightarrow \left \{ -1,1 \right \} $ implies
$ \left | \...
3
votes
0
answers
131
views
Chen primes and permutations
In 1973 the Chinese mathematician J.-R. Chen proved that there are infinitely many primes $p$ such that $p+2$ is a product of at most two primes. Nowadays such primes $p$ are called Chen primes.
For $...
- 15.6k
4
votes
2
answers
492
views
Distance between primes that are quadratic residues modulo an other prime
Question: Is there an infinite sequence of primes $\{q_i\}_{i=1}^{\infty}$ that is not too sparse ( $q_n =O(poly(n))$ for a fixed polynomial) for which it is true that for every $k$ there is an $N(k)$ ...
- 3,004
8
votes
2
answers
1k
views
Prime plus square equals prime
Furstenberg–Sárközy's theorem states that if a set of positive integers has positive upper density, then there exists (infinitely many) pair of elements of the set, whose difference is a perfect ...
- 191
5
votes
0
answers
355
views
What is the sum of the binomial coefficients ${n\choose p}$ over prime numbers?
What is known about the asymptotics, lower and upper bound of the sum of the binomial coefficients
$$
S_n = {n\choose 2} + {n\choose 3} + {n\choose 5} + \cdots + {n\choose p}
$$
where the sum runs ...
- 5,470
5
votes
1
answer
2k
views
Is there a relation between the number of lattice points lie within these circles
Suppose we have a circle of radius $r$ centered at the origin $(0,0)$. The number of integer lattice points within the circle, $N$, can be bounded using Gauss circle problem.
Suppose that another ...
- 225
37
votes
3
answers
2k
views
How to prove the identity $L(2,(\frac{\cdot}3))=\frac2{15}\sum\limits_{k=1}^\infty\frac{48^k}{k(2k-1)\binom{4k}{2k}\binom{2k}k}$?
For the Dirichlet character $\chi(a)=(\frac a3)$ (which is the Legendre symbol), we have
$$L(2,\chi)=\sum_{n=1}^\infty\frac{(\frac n3)}{n^2}=0.781302412896486296867187429624\ldots.$$
Note that this ...
- 15.6k
4
votes
1
answer
314
views
Asymptotic for number of partitions of $n$ into $k$ squares, uniform in $n,k \to +\infty$
Let $p^{(s)}(n)$ be the number of ways of writing the positive integer $n$ as a sum of perfect $s$-powers, where the order does not matter. For example, $p^{(2)}(9) = 4$ since
$$9 = 1^2 + 1^2 + 1^2 + ...
- 41
6
votes
2
answers
385
views
Is the nth-power-sum graph connected?
This post was inspired by the Square-Sum Problem presented in Numberphile by Matt Parker.
He asked about Hamiltonianness for $n=2$, and we ask about connectedness for all $n \in \mathbb{N}^*$.
...
0
votes
1
answer
132
views
Divisibility criterion of binomial coefficients
If $r\in\Bbb Z_{\geq0}$ and $m$ is odd then let $2^\ell\mid\binom{m}{2^r}$ and $2^{\ell+1}\nmid\binom{m}{2^r}$.
Is there a way to find if $\ell$ is even or odd without computing $\binom{m}{2^r}$ (...
- 13.9k
10
votes
3
answers
1k
views
Explicit formula for elementary symmetric sum
For $k\ge1$, $j\ge1$, Let $$e_k(j)=\sum_{1\le i_1<...<i_k\le j}i_1\cdot\cdot\cdot i_k.$$ We know that $e_k(j)$ is a polynomial in $j$ with coefficients depending on $k$. I am curious about ...
- 113
6
votes
1
answer
305
views
Are the Fourier coefficients of $\eta(q^m)^m / \eta(q)$ non-negative?
In this paper, the following result is proved.
For any prime $p$, all the Fourier coefficients of
$$\eta(q^p)^p / \eta(q) = q^{\frac{p^2-1}{12}} \prod_{n=1}^\infty (1 - q^{pn})^p (1 - q^{n})^{-1}$$
...
- 427
2
votes
1
answer
423
views
Essential clarifications on application of pigeonhole principle
In here Lemma $4$ using pigeonhole says:
For $T_1,\dots,T_s\in\Bbb R$ with $1\leq T_1,\dots,T_s<p$ and $\prod_{i=1}^sT_i > p^{s−1}$ and any integers $a_1,\dots,a_s$ there is an integer $t$ ...
user94040
4
votes
1
answer
464
views
Odd Chebyshev, part 2
Let
$$ \forall_{n=1\ 2\ \ldots}\quad I(n)\ :=\
\frac {(6\cdot n-3)!!}{(2\cdot n-1)!!\cdot(4\cdot n-3)!!} $$
Let $\ M(n)\ $ be the smallest natural number such that
$$ M(n)\cdot I(n)\ \...
- 7,500
5
votes
0
answers
170
views
operation on Ord., Exp., Dri. generating functions
The ordinary, exponential and Dirichlet generating functions for a sequence $\{a_n\}_{n\geq0}$ are given (at least on the formal side), respectively, by
$$F(x)=\sum_{n\geq0}a_nx^n, \qquad E(x)=\sum_{n\...
- 43.2k
1
vote
0
answers
118
views
Carry operations when adding two numbers [closed]
Let $x$ be a large positive real number and let $q\leq 2$ be a positive integer. It is known that for a positive integer $n,$ there exists a unique sequence $\left\{0\leq n_k\leq q-1\right\}_{k\geq 0}$...
- 1,257
3
votes
0
answers
320
views
On sets of coprime numbers
We know that from prime number theorem that the number of primes below $n$ and above $\frac n2$ (denoted by $\pi_{n,\frac n2}$ is approximately $$\pi_{n,\frac n2}\approx\frac{n}{2\ln n}.$$
Denote by $...
user76479