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Results tagged with prime-numbers
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user 10898
Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.
4
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0
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398
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Efficient ways to count primes satisfying Zhang's theorem
The theorem of Yitang Zhang states that there exist a finite $k \in \mathbb{N}$ such that there exist infinitely pairs of primes $(p,q)$ such that $|p - q| \leq k$. The statement that $k$ can be taken …
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2
votes
1
answer
357
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Generalizations of Chen's theorem
The two famous theorems of Jingrun Chen, both with similar proofs, state (respectively) that all sufficiently large even numbers are the sum of a prime and an element of $P_2$, and that there are infi …
- 26.9k
1
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0
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117
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For which types of problems can one expect to use Bombieri-Vinogradov in place of GRH?
I must profess a general ignorance of problems that were once known to be true under GRH but has since became unconditional due to the Bombieri-Vinogradov theorem, but I am aware of the heuristic that …
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4
votes
0
answers
241
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Can the following quantitative version of Chen's theorem be obtained?
The theorem of the Chinese mathematician Chen Jingrun is currently one of the best results with respect to the binary Goldbach problem. It asserts that every even integer $n$ is in the sumset $P + P_2 …
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9
votes
1
answer
1k
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A general question about strictly non-palindromic numbers
For a definition, see the wikipedia page: http://en.wikipedia.org/wiki/Strictly_non-palindromic_number
So according to the wikipedia page, under properties, all strictly non-palindromic numbers with …
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10
votes
4
answers
1k
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A question about primes as an additive basis
Let $\mathcal{P}$ denote the set of primes. Define the function $r_2(N)$ to be the number of ways to write $N$ as a sum of two not necessarily distinct primes (where order matters). Then the famous Go …
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3
votes
1
answer
452
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An estimate for 'almost primes'?
In the famous Chen's Theorem which states that every sufficiently large even positive integer $n$ can be written as $n = p + q$, where $p$ is a prime and $q$ is a product of at most two primes. This i …
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7
votes
1
answer
481
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Higher dimensional generalization of the Hardy-Littlewood conjecture?
The famous Hardy-Littlewood conjecture on prime-tuples states that if $\{h_1, \cdots, h_k\} = \mathcal{H}$ is an admissible set, that is, for every prime $p$ the set $\mathcal{H}$ does not contain a c …
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3
votes
0
answers
321
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Density of numbers whose prime factors all come from a fixed congruence class
Let $q$ be a positive integer greater than one, and let $a$ be an integer such that $\gcd(a,q) = 1$. Define
$$D(a,q) = \{n \in \mathbb{N} : p | n \Rightarrow p \equiv a \pmod{q} \}.$$
Do we know the d …
- 26.9k
4
votes
0
answers
262
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A variant of the Green-Tao theorem
Green and Tao famously proved (The primes contain arbitrarily long arithmetic progressions) that there are arbitrarily long arithmetic progressions in the primes. Specifically, for $k = 3$ this implie …
- 26.9k
11
votes
1
answer
602
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Mersenne almost primes
I asked earlier whether it can be proved that infinitely many elements of $P_n$ for some positive value of $n$ (here $P_n$ refers to the set of numbers with at most $n$ prime divisors). There I receiv …
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2
votes
1
answer
156
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Primality of divisor sums
Let $k \geq 2$ be an integer. Put $[k] = \{1, \cdots, k\}$. Let $\mathcal{P} = \{p_1, \cdots, p_k\}$ be a set of $k$ primes. For every subset $S \subseteq [k]$ put $d_S = \prod_{j \in S} p_j$. The emp …
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0
votes
0
answers
162
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Sequences that "capture their primes"
Let $g : \mathbb{N} \rightarrow \mathbb{N}$ be an increasing function, and consider the sequence $Y = \{y_n\}$ given by $y_n = g(n)$. Let $F$ be an irreducible binary form of degree $d \geq 2$ with in …
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12
votes
2
answers
614
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Are there any notion of 'almost primes' known to have small gaps?
A notorious question with prime numbers is estimating the gaps between consecutive primes. That is, if $(p_n)_{n \geq 1}$ is the canonical enumeration of the primes, then set $g_n = p_{n+1} - p_n$. It …
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5
votes
1
answer
387
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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 …
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