All Questions
26 questions
24
votes
3
answers
2k
views
Are sets with similar asymptotic behavior as the primes necessarily finite additive bases?
The set of primes $\mathbb{P}$ has many interesting properties in additive number theory and some of the most famous open problems about $\mathbb{P}$ are the well-known Goldbach's strong and weak ...
- 2,719
21
votes
1
answer
773
views
Avoiding multiples of $p$
Let $p$ be a prime number and $P=\{1,2,...,p-1\}$
In how many ways we can sum all the elements of $P$ in such a way that we will reach a multiple of $p$
only when we sum the last summand?
For ...
- 3,866
11
votes
2
answers
1k
views
Most dense subset of numbers that avoids arbitrarily long arithmetic progressions
The famous Green-Tao theorem says that there exist arbitrarily long sequences of primes in arithmetic progression.
I am wondering: How dense can a subset $S \subset \mathbb{N}$ be and still avoid
...
- 151k
6
votes
3
answers
775
views
A simple looking problem in partitions that became increasingly complex
I began with problem which looked simple in the beginning but became increasingly complex as I dug deeper.
Main questions: Find the number of solutions $s(n)$ of the equation
$$
n = \frac{k_1}{1} + \...
- 5,470
5
votes
3
answers
2k
views
Goldbach conjecture and other problems in additive combinatorics
The field is also known as additive number theory. I am interested in sums $z=x + y$ where $x \in S, y\in T$, and both $S, T$ are infinite sets of positive integers. For instance:
$S = T$ is the set ...
- 3,098
5
votes
2
answers
691
views
Representing natural numbers as sums of distinct prime powers
I am investigating whether every natural number $n > 18$ can be represented as a sum $p_1^{m_1} + \dots + p_k^{m_k}$, where $p_1, \dots, p_k$ are distinct primes, and $m_1, \dots, m_k$ are distinct ...
- 1,228
5
votes
1
answer
351
views
Is every integer $\ge 312$ the sum of two integers with triangular divisors?
We say that a natural number $n$ has triangular divisors if it has at least one triplet of divisors $n = d_1d_2d_3$, $1 \le d_1 \le d_2 \le d_3$, such that $d_1,d_2$ and $d_3$ form the sides of a ...
- 5,470
5
votes
1
answer
389
views
Thin subbases for the primes?
Hi all,
My question concerns a general problem concern the Erdos-Turan conjecture on additive bases; that of finding thin subbases in a given basis. For a given $A \subset \mathbb{N}$, define $r_{A,h}...
- 26.9k
5
votes
0
answers
83
views
Maximum size of difference sets with a bounded number of prime divisors
Call a subset $S\subset \mathbb{Z}$ $r$-smooth if the difference set $S-S$ contains numbers whose prime divisors lie in a set $P$ of distinct primes with $|P|=r$. Let $f(r)$ be the maximum size of any ...
- 4,862
3
votes
2
answers
481
views
Two equivalent statements about primes
Regarding to our hypothesis in https://math.stackexchange.com/questions/1918406/a-hypothesis-about-the-conjecture-every-even-number-is-the-difference-of-two-p , we guess that the following statements ...
- 995
3
votes
0
answers
165
views
What is the density of numbers which have at least two divisors whose sum is a perfect square?
Note: This question was posted in MSE about two years ago but it not receive an answer. Hence posting in MO.
A positive integer is said to have square-sum divisors if it has at least two divisors ...
- 5,470
3
votes
1
answer
360
views
Prime gap distribution in residue classes and Goldbach-type conjectures
Update on 7/20/2020: It appears that conjecture A is not correct, you need more conditions for it to be true. See here (an answer to a previous MO question).
The general problem that I try to solve is ...
- 3,098
3
votes
0
answers
265
views
Prove A Skipping Prime Conjecture For Rio?
I am writing a paper to accompany a Short Communication I plan to give in Rio this August. The paper regards work on jumping primes, a project on which Jose Brox has been working with me. I was going ...
- 13k
2
votes
1
answer
617
views
every arithmetic progression contains a sequence of $k$ "consecutive" primes for possibly all natural numbers $k$?
I ask the same question here:https://math.stackexchange.com/q/1019404/192097
writing a little better the previous question: it´s true that if we let $a$ and $b$ be coprime integers, then the ...
- 21
2
votes
2
answers
393
views
Playing leapfrog with primes
In connection with how primes jump (How do these primes jump?),
I consider the following game.
Let $R$ be a finite set of positive integers. For this question, I content myself with $R$ being the $k$ ...
- 13k
2
votes
0
answers
140
views
Primality radii and Sidon sets
I learned tonight what a Sidon set is, in a book about Erdős. This notion inspires me the following question :
For $n$ a large enough composite integer, say $r>0$ is a primality radius of $n$ if ...
- 6,984
2
votes
0
answers
292
views
Prime divisors of the difference set
Fix $c\in(0,1)$, and let $N$ be a (large) positive integer. Given a set $A=\{0=a_1<\dots<a_n=N\}$ of density $\alpha:=n/N>c$ with $\gcd(A)=1$, I want to find a prime dividing as few ...
- 23k
1
vote
0
answers
55
views
Largest interval containing family of sets with an overlap property
Here's a simplified version of a question I'm interested in.
Given $p$ and $q$ distinct prime numbers, we consider sets $A\subset \mathbb{N}\cup\{0\}, 0\in A$ of size $pq$, which are uniformly ...
- 549
1
vote
0
answers
98
views
Reference request for a result in additive combinatorics
Let $p$ be a prime number and $[p-1]=\{1, 2, \ldots, p-1\}$.
The following proposition is proved: (but I cannot find out where)
Proposition: The non-empty subset sums of $[p-1]$ are equally ...
- 3,866
1
vote
0
answers
141
views
On certain number theoretic sextuples?
Given small parameters $0<\epsilon<\epsilon'$ is there an $n_\epsilon>0$ such that at every $n>n_\epsilon$ if we are given a prime $n^2<p<2n^2$ then can we always find integers $a,b,...
- 13.9k
1
vote
0
answers
165
views
Are the Beatty primes asymptotically (Gowers) uniform?
A result of Green and Tao (initially conditional on two conjectures which were eventually settled by them and Ziegler) states that for any $s\in\mathbb N$,
$$\lim_{w\to\infty}\limsup_{N\to\infty}\sup_{...
- 1,701
1
vote
0
answers
402
views
Green-Tao style theorem for quadratic regressions (Ulam Spiral)
This is a naive question about number theory.
Looking at an Ulam spiral which illustrates primes of the form e.g. $4x^2-2x+c$ and other quadratic equations $ax^2+bx+c$, with $c>0$, there appears a ...
- 1,066
0
votes
1
answer
489
views
Congruential equidistribution, prime numbers, and Goldbach conjecture
Let $S$ be an infinite set of positive integers, $N_S(z)$ be the number of elements of $S$ less than or equal to $z$, and let
$$D_S(z, n, p)= \sum_{k\in S,k\leq z}\chi(k\equiv p\bmod{n}).$$
Here $\chi$...
- 3,098
0
votes
1
answer
669
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 ...
- 3,098
0
votes
0
answers
374
views
Is the Conjecture of Representing Integers as Differences of Semiprimes and Primes Extendable to Products of Distinct Primes?
Conjecture:
Let $k$ and $l$ be fixed distinct positive integers ($k≠l$). Then, for every positive integer $n$, there exist prime numbers $p_1,p_2,…,p_k∈\mathbb{P}$ and $q_1,q_2,…,q_l∈\mathbb{P}$ such ...
- 17
0
votes
0
answers
58
views
Existence of minimal bases in additive combinatorics
Let $\mathbb{N}$ denote the set of natural numbers, including zero. A subset $X \subseteq N$ is a basis if $X + X = \mathbb{N}$. Clearly, if $X$ is a basis and $X \subseteq Y$, then $Y$ is also a ...
