Questions tagged [sieve-theory]

Sieve theory deals with number theoretic sieves, and sifted sets. E.g. the Sieve of Eratosthenes, Brun sieve, and other modern sieves.

260 questions
36
votes
0 answers

Using the Brun Sieve to show very weak approximation to twin prime conjecture

I recently stumbled across MIT OCW for analytic number theory. As a budding number theorist, my ears perked up and I looked through some of the material haphazardly. I don't really know much about sieves, but I became intrigued when I read over the…
davidlowryduda
  • 86,205
  • 9
  • 157
  • 296
16
votes
0 answers

How does a Lehmer Sieve work?

http://en.wikipedia.org/wiki/Lehmer_sieve Apparently a Lehmer Sieve was a mechanical device that used chains and pulleys to factor numbers and solve diophantine equations. It once was able to factor $2^{93} + 1 = 3 \times 3 \times 529510939 \times…
15
votes
2 answers

Most efficient algorithm for nth prime, deterministic and probabilistic?

What's the most efficient algorithm for calculating an $nth$ prime, both deterministically and probabilistically? Deterministic Iterate through only odd values, incrementing by $2$. Divide each value by $2 < divisor < \sqrt{value}$, where…
13
votes
0 answers

Twin-prime sieve

My question concerns the following sieve (call it S), which was an exercise in applying some elementary aspects of Brun's sieve while reading Halberstam's text. Using the Chinese Remainder theorem we can show: (S) The number of $n$ such that…
daniel
  • 9,714
  • 5
  • 30
  • 66
12
votes
1 answer

primegaps w.r.t. the m first primes / jacobsthal's function

Maybe I don't see the obvious here; but well. I looked at an old discussion concerning prime gaps. I now tried to ask a somehow opposite way: Assume the set $\small P(m)$ of first m primes $\small \{p_1,p_2, \ldots,p_m\}$ . Then consider…
Gottfried Helms
  • 32,738
  • 3
  • 60
  • 134
11
votes
4 answers

Why in Sieve of Erastothenes of $N$ number you need to check and cross out numbers up to $\sqrt{N}$? How it's proved?

Why in Sieve of Erastothenes of $N$ number you need to check and cross out numbers up to $\sqrt{N}$? How it's proved? For example if $N = 20$: with $2$ we cross out: 2 4 6 8 10 12 14 16 18 20 with $3$: 3 9 15 and with $5$ we don't need to check…
Templar
  • 1,643
  • 3
  • 22
  • 36
11
votes
3 answers

Gaussian Primes

I need to adapt the Sieve of Eratosthenes for the usual integers to find all Gaussian primes with norm less than a specific limit. How to apply it to finding all Gaussian primes with norm less than 100? Thank you very much!
user9636
  • 301
  • 1
  • 11
11
votes
3 answers

How to tell if a particular number will survive in this sieve?

I was asked this in an interview. We have people numbered from one to infinity: $$1, 2, 3, 4, 5, 6, 7, 8, \dotsc\,.$$ In first pass every 2nd person is killed, so we have $$1, 3, 5, 7, 9, 11,\dotsc$$ remaining. In next pass every 3rd remaining…
Oliver Blue
  • 365
  • 2
  • 10
11
votes
2 answers

On the Cramér-Granville Conjecture and finding prime pairs whose difference is 666

Questions If $p= \text{NextPrime}[q]$ (the smallest prime greater than $p$), and $p-q = 666,$ what are $p$ and $q$? (There may be multiple choices. I am interested in finding one.) Cramér-Granville Conjecture: Defining $p_0=2$, and $p_n$ as the…
10
votes
2 answers

Accuracy of approximation to inclusion-exclusion formula in prime sieve

This thing came up in a combinatorics course I am taking. Choose a fixed set of primes $p_1,p_2,\dots,p_k$ and let $A_n$ be number of integers in $\{1,2,\dots,n\}$ which are not divisible by any of the $p_i$'s. $A_n$ is given by $ n - \sum_{1\leq…
9
votes
2 answers

"Dirichlet's theorem" on pairs of consecutive primes

The number of primes in each of the $\phi(n)$ residue classes relatively prime to $n$ are known to occur with asymptotically equal frequency (following from the proof of the Prime Number Theorem). Does the same result hold on pairs of consecutive…
Charles
  • 30,888
  • 4
  • 58
  • 139
9
votes
2 answers

Prove there are infinitely many primes in $\mathbb{Z}[i]$

I saw a proof online there are infinitely many primes in $\mathbb{Z}$. The Euler product let's us factor the harmonic series: $$ \prod \left( 1 - \frac{1}{p} \right) = \sum \frac{1}{n}$$ I wonder if this extends to $\mathbb{Z}[i]$. Joan Baez Week…
cactus314
  • 23,583
  • 4
  • 35
  • 88
7
votes
1 answer

Are there infinitely many primes that are 1 more a square-free number?

Other words, are there infinitely many primes $p$ such that $p-1$ is square-free. This seems likely to be true, but I can't seem to give an easy argument why. This set of primes and all primes 1 more an integer with a square divisor make up the set…
7
votes
4 answers

Computing the first $n$ values of the Liouville function in linear time

Is it possible to compute the first $n$ values of the Liouville function in linear time? Since we need to output $n$ values we clearly cannot do better than linear time, but the best I can figure out is something like $O(n \cdot \log{\log{n}})$:…
7
votes
1 answer

parity problems for sieve methods, is it only for Selberg Sieve or for all sieve methods?

It is said that sieve methods have parity problems. Terence Tao gave this "rough" statement of the problem: "Parity problem. If A is a set whose elements are all products of an odd number of primes (or are all products of an even number of primes),…
1
2 3
17 18