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Results tagged with prime-numbers
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user 2384
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.
14
votes
Accepted
form of primes:prime plus a power of 2?
127 and 331 are counterexamples. It was a conjecture of Polignac that every odd number can be written as a sum of an odd prime and a power of two, but many counterexamples have been found. They are ca …
- 85.6k
4
votes
Smoothness in Mersenne numbers?
I guess lower bounds on the largest prime factor of Mersenne numbers are not only interesting in number theory but also in coding theory (see this article of K. Kedlaya and S. Yekhanin here). They say …
- 85.6k
9
votes
Accepted
Subsets of $\mathbb N$ with a finite number of prime factors
Proving that $a_{k+1}-a_k\to \infty$ reduces to proving that $$a_{k+1}-a_k=n$$ has finitely many solutions for every $n$. If you let $S$ be the set of prime divisors of $n$ union $P$, then this follow …
- 85.6k
4
votes
Accepted
An abelian group associated to divisors of an integer $N$
All the $\tau$'s fix $0$ so we can look at how they behave on the set $\{1,2,\dots,N-1\}$. We have
$$\tau_{d|N}(kd+r)=k+\frac{rN}{d}=\left[\frac{N}{d}(kd+r)\right]_{\pmod{N-1}}$$
where $[a]_{\pmod{N-1 …
- 85.6k
10
votes
Primes $P_{2n-1}$ that are $2$ mod $3$
It was known since Littlewood, "Distribution des Nombres Premiers", C. R. Acad. Sci. Paris 158 (1914), 1869-1872, that there are infinitely many $x$ for which $\pi(x,3,1)>\pi(x,3,2)$ as well as infini …
- 85.6k
11
votes
Accepted
Computing the Mertens function
This article presents an algorithm to compute Mertens function in $O(x^{2/3}(\log \log x)^{1/3})$ time and $O(x^{1/3}(\log \log x)^{2/3})$ space, I wonder if it is the same one you are referring to. O …
- 85.6k
6
votes
Is $n = p-q$ equivalent to Goldbach's Conjecture?
Close to nothing can be said rigorously about your question, but I believe the following heuristics:
It's hard to imagine the exact solution of either conjecture not leading to substantial progress …
- 85.6k
8
votes
Accepted
Primes in quasi-arithmetic progressions?
I think the uniform distribution mod1 of $\{p/\alpha\}$ is due to Vinogradov, and the asmptotic for primes in a Beatty sequence $\sim \frac{\pi(x)}{\alpha}$ is an immediate consequence. Indeed for $p$ …
- 85.6k
14
votes
Accepted
Recovering n from sigma(n)/n
For the first question, note that the set $\Delta=\operatorname{im} (\delta)$ is not known very well understood. For example $5/3\in \Delta$ implies the existence of an odd perfect number. (C.W. Ander …
- 85.6k
6
votes
Accepted
Multiplicity one prime in the factorisation of p-N
The number of primes in $[N,3N/2]$ grows as $\frac{N}{\log N}$, while the number of powerful numbers in $[1,N/2]$ grows as $\sqrt{N}$, so pretty quickly you will find primes $p\in [N,3N/2]$ so that $p …
- 85.6k
16
votes
Generalization of Tamarkin's ARO 1993, final round, problem 10/8: still a conjecture?
This problem is a lot of fun! There is a way you can reduce the general problem to studying average-integral polynomials (take an average-integral sequence, pick a finite but large enough subsequence, …
- 85.6k
37
votes
Accepted
Is this BBP-type formula for $\ln 257$ and $\ln 65537$ true?
Yes, this actually holds for all Fermat numbers! Let's start with the identity
$$\log 2=\sum_{k=1}^{\infty}\frac{1}{k2^k}$$
and try to work out an expression for
$$\log (2^{2^s}+1)=2^{s}\log 2+\log\l …
- 85.6k
16
votes
Accepted
Walking to infinity on the primes: The prime-spiral moat problem
I don't know if any of the probabilistic or percolation models related to the Gaussian (Eisenstein etc.) prime walks have analogues for the problem you suggest. However note that if such an infinite w …
- 85.6k
33
votes
Accepted
Injective integer polynomial is injective modulo some prime
Consider $Q(x)=x(2x-1)(3x-1)$. This gives an injective map $\mathbb Z\to \mathbb Z$, because $n<m \implies Q(n)<Q(m)$. However, this $Q$ is not injective over $\mathbb Z/p\mathbb Z$ for any $p$ becaus …
- 85.6k
62
votes
Accepted
What are the connections between pi and prime numbers?
Well, first of all, $\pi$ is not just a random real number. Almost every real number is transcendental so how can we make the notion "$\pi$ is special" (in a number-theoretical sense) more precise?
St …