Skip to main content

All Questions

Filter by
Sorted by
Tagged with
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 ...
Akira Sukigi's user avatar
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 ...
Marcos Cramer's user avatar
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 ...
Pathikrit Basu's user avatar
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 ...
Itay's user avatar
  • 549
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 ...
Nilotpal Kanti Sinha's user avatar
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 ...
Konstantinos Gaitanas's user avatar
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 ...
Ivan Meir's user avatar
  • 4,862
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 ...
Vincent Granville's user avatar
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 $\chi$...
Vincent Granville's user avatar
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 ...
Vincent Granville's user avatar
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 ...
Vincent Granville's user avatar
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 ...
Nilotpal Kanti Sinha's user avatar
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 ...
Sylvain JULIEN's user avatar
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 ...
Gerhard Paseman's user avatar
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,...
Turbo's user avatar
  • 13.9k
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 $k$ ...
Gerhard Paseman's user avatar
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 ...
M.H.Hooshmand's user avatar
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_{...
Joel Moreira's user avatar
  • 1,701
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 ...
Joseph O'Rourke's user avatar
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 ...
Cristhofer's user avatar
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 ...
Konstantinos Gaitanas's user avatar
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 ...
Zahlendreher's user avatar
  • 1,066
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} + \...
Nilotpal Kanti Sinha's user avatar
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 ...
Joni Teräväinen's user avatar
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 ...
Seva's user avatar
  • 23k
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}...
Stanley Yao Xiao's user avatar