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Results tagged with prime-numbers
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user 307675
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
votes
Does there exist an even positive integer $n$ such that, for each prime number $p>2$, $p+n$ ...
The answer is no, if you believe the Hardy-Littlewood $k$-Tuple Conjecture. Let $\pi_{k}(x)$ denote the number of primes $p\leq x$ such that $p+2k$ is also prime. Then the conjecture predicts
$$
\pi_{ …
- 1,990
5
votes
Accepted
Upper bound of number of prime factors
To reiterate Stanley Yao Xiao's comment, one should not expect that $p-1$ has any more or less prime factors than a typical integer of size $p$. For an arbitrary integer $n$, the bound
$$
\omega(n) \l …
- 1,990
13
votes
About an asymptotic behavior in number theory
See this paper of Naslund, specifically Proposition 3. That result, along with the prime number theorem, shows that
$$
\frac{1}{\pi(N)} \sum_{p\leq N} \left\{ \frac{N}{p} \right\} \sim 1-\gamma,
$$
wh …
- 1,990
6
votes
Approximation of partial sum over prime omega function
By partial summation, one has
$$
S(N) = N\sum_{n=1}^N \omega(n) - \int_{1}^N \bigg(\sum_{n\leq t} \omega(n)\bigg) dt.
$$
Using Mertens' theorem with the classical error term in the prime number theore …
- 1,990
6
votes
0
answers
148
views
Dickson's conjecture for Beatty sequences
A particular case of Dickson's Conjecture states that for $a_1,q_1,a_2,q_2$ with $(a_1,q_1)=(a_2,q_2)=1$, there are infinitely many $n$ for which $q_1 n + a_1$ and $q_2 n+a_2$ are both prime, provided …
- 1,990
1
vote
Non-trivial upper bound for a sum related to $p^{-1}z \pmod q$ and $q^{-1}z \pmod p$
This is not mean to be a full answer, but one which illustrates how one can prove an estimate like the one in my comment through a rather ``brute force'' approach.
To illustrate the idea of the comput …
- 1,990
5
votes
1
answer
213
views
Remainder terms of congruence sums in sets of positive density
Let $\mathcal{A} \subset \mathbb{N}$ be an infinite sequence with positive density, in the sense that
$$
\tag{1}
\lim_{x\to\infty} \frac{|\mathcal{A} \cap x|}{x} = c > 0,
$$
and define the remainder t …
- 1,990