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Results tagged with prime-numbers
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user 6043
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.
3
votes
0
answers
1k
views
Generating random smooth numbers (or, "What do random smooth numbers look like?")
What is a good method for generating random b-bit, S-smooth numbers?
For S large and b not too large, it may be feasible to generate random numbers and test if they are smooth enough. If S is too sma …
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1
vote
1
answer
305
views
Interpreting a paper: primes and interval size
I was reading the Polymath4 project and have trouble understanding one of the arguments. From page 6 of [1] (either the preprint or the final paper):
For any j ≥ 2, the interval $[a^{1/j}, b^{1/j …
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3
votes
1
answer
988
views
Whence the k-tuple conjecture?
What is the source of the $k$-tuple conjecture, that every integer tuple $(k_1,\ldots,k_n)$ either contains all members of a congruence class mod a prime or has infinitely many primes amongst $(k_1+c, …
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1
vote
1
answer
452
views
Are large numbers the sum of two or more large primes? [Hoping for reasonable constants]
Is it true that for all $n>N$ that n is the sum of two or more distinct primes that are either large or (for parity reasons) 2?
I feel like I've seen a result allowing this with $p\gg n^e$ for reason …
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3
votes
0
answers
1k
views
Effective upper bound on large prime gaps; or, what is the first prime after a googolplex?
Question
What is the best known effective upper bound on the prime gap following x?
Motivation
Suppose you needed to show a good bound for the gap between a fixed large constant, say $G=10^{10^{100 …
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5
votes
4
answers
785
views
Proving a least prime factor
Suppose that I find a small prime factor $p$ dividing a large number $n$ and I wish to prove that it is the least prime dividing $n$. There are two obvious approaches: either factor $n/p$, or divide $ …
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29
votes
7
answers
7k
views
Asymptotic density of k-almost primes
Let $\pi_k(x)=|\{n\le x:n=p_1p_2\cdots p_k\}|$ be the counting function for the k-almost primes, generalizing $\pi(x)=\pi_1(x)$. A result of Landau is
$$\pi_k(x)\sim\frac{x(\log\log x)^{k-1}}{(k-1)!\l …
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1
vote
0
answers
301
views
Is this a proof of the Hardy-Littlewood inequality? [closed]
V.V. Miasoyedov posted a paper to the arXiv claiming a proof of the Hardy-Littlewood conjecture $\pi(x+y) \le \pi(x)+\pi(y)$. It seems a bit off, and not only because the conjecture is widely believed …
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2
votes
0
answers
149
views
$f(x)$-th largest number of prime factors
Given a sufficiently well-behaved function $1\le f(x)\le x$ and a multiset $S=\{\omega(n): 1\le n\le x\}$, what can be said about the asymptotics of the $f(x)$-th largest member of $S$? In other words …
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4
votes
2
answers
1k
views
Product over the primes
I'm trying to estimate the product
$$\prod_{p\lt q\lt r\lt s}1-\frac{24}{(pqrs)^2}$$
where $p,q,r,s$ are primes.
This is for the purpose of calculating the density of Sloane's A070284 [1]. The idea i …
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10
votes
2
answers
3k
views
Can a number be factored quickly, given the sum of its prime factors?
This is perhaps most naturally phrased as a promise problem. Given numbers $n$ and $s$, where $s$ is the sum of the prime factors of $n$ (distinct or with multiplicity; I imagine both variants will ha …
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12
votes
1
answer
864
views
Analytic lower bounds on the first sign change of pi(x) - li(x)?
There have been many results on the first sign change of $\pi(x)-{\mathrm{li}}(x)$: among others, Lehman, te Riele, Bays & Hudson, Demichael, Chao & Plymen, and most recently Saouter & Demichel. Thes …
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4
votes
1
answer
491
views
Can we count primes in residue classes quickly?
Using combinatorial methods (due to Legendre, Lehmer, Meissel, Lagarias, Miller, Odlyzko, Deléglise, Rivat, and probably others) it's possible to count the number of primes up to $N$ quickly -- in tim …
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7
votes
1
answer
421
views
Are primes of density 0 in $a\cdot b^n+c$?
Hooley proves in Applications of Sieves to the Theory of Numbers that there are only $o(x)$ numbers $n\le x$ such that $n\cdot2^n+1$ is a (Cullen) prime. The proof generalizes to forms $n\cdot2^{n+a}+ …
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38
votes
4
answers
7k
views
What did Yu Jianchun discover about Carmichael numbers?
There's a news story going around (see for example [1]; other accounts are even more breathless) about an amateur mathematician, Yu Jianchun, finding an "alternative method to verify Carmichael number …
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