All Questions
9 questions
3
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Roth's theorem for primes in a given arithmetic progression to a large modulus
Let $\mathbb{P}_{a, q}$ denote the set of primes congruent to $a$ modulo $q$. Are there any estimates for the number of $3$-Arithmetic Progressions in the set $\mathbb{P}_{a, q}\cap [1, X]$, where $(a,...
- 31
3
votes
1
answer
220
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Bins-and-primes (prime divisors of $\prod(a_i-a_j)$, II)
This is a refined version of a question I have recently posted.
For a prime $p$, let $\varphi_p\colon\mathbb Z\to\mathbb Z/p\mathbb Z$ denote the canonical homomorphism from the integers onto the ...
- 23k
2
votes
1
answer
163
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Prime divisors of $\prod(a_i-a_j)$
For a prime $p$, let $\varphi_p\colon\mathbb Z\to\mathbb Z/p\mathbb Z$ denote the canonical homomorphism from the integers onto the group of order $p$.
Given an integer $n\ge 3$, what is the smallest ...
- 23k
3
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0
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165
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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
5
votes
3
answers
2k
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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
1
answer
351
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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
1
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0
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141
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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
3
votes
2
answers
481
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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
2
votes
0
answers
292
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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