0
votes
2answers
140 views

Twin Prime Conjecture's Proof [closed]

I've found this article that claims to have a proof of the Twin Prime Conjecture. Can you find any error? (I have some doubts about the last page of the paper...)
3
votes
1answer
126 views

Explanation for Terry T. post

I read here that : " If one inserts these inequalities into the Legendre sieve and optimises the parameter, one can improve the upper bound for the number of primes in $[N,2N]$ to $$O \left(\frac{N ...
2
votes
1answer
122 views

Sieving integers

Let $2=p_1,p_2,\cdots ,p_n$ be the first $n$ prime numbers. Suppose $N$ is a natural number and that $A=\{a+1,\cdots, a+N\}$ be a set of $N$ consecutive integers. Let $P_n=p_1\cdot p_2\cdot \cdots ...
0
votes
0answers
51 views

Recommend a good read on prime sieving for primes of the form $x^2 + y^2$

Recommend a good read on prime sieving, which can be applied to sieve primes of the form $x^2 + y^2$ (or $x^2 + ny^2$, if possible). I actually need to find up to a certain limit numbers that are ...
1
vote
0answers
77 views

Sieve for Prime Numbers

I will use some simple arguments on a prime numbers formula that has been deterministically checked by computer. I would like to compare this result with others you already know. The set of all ...
2
votes
0answers
40 views

Can anyone sketch an outline of Iwaniec's proof for the upper bound regarding the Jacobsthal function?

A proof by H. Iwaniec in 'On the problem of Jacobsthal, Demonstratio Math. 11, 225–231, (1978)' shows that: $$j(N) \ll \log \log (N)$$ where $j(N)$ is the Jacobsthal function. I am very interested ...
4
votes
1answer
100 views

Are most numbers of the form $a\cdot b^n+c$ composite?

It seems evident that for $a,b,c$ with $a>0$ and $b>1$ that there are only $o(x)$ primes of the form $a\cdot b^n+c$ with $n\le x.$ Has this been proven? Hooley (Applications of Sieves to the ...
0
votes
0answers
76 views

Is my sieve generalisable?

I was curious about extending Euler's polynomial generator n^2 - n + 41 for n > 41, and looking for the simplest sieves. I examined the gaps between non-primes and found a set of simple sieves of the ...
1
vote
1answer
561 views

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 ...
5
votes
1answer
80 views

Counting numbers of the form $ai + bj + cij$ and finding related L-series?

Let $a,b,c$ be given nonnegative integers with $gcd(a,b,c)=1$. Consider a given positive integer $n$ and positive integers $i,j$. Let $f_n(a,b,c)$ be the number of distinct solutions to $1<ai + bj ...
0
votes
2answers
48 views

At what rate are composites removed in a set after each prime multiple is cancelled out?

I was looking at sieves today, mainly sieving for primes and I noticed a pattern type thing. As I crossed out primes in a small set, the number of composites that were crossed out decreased. I haven't ...
2
votes
0answers
278 views

Confusing proof of brun's theorem?

I read Brun's proof of Brun's theorem here : http://gallica.bnf.fr/ark:/12148/bpt6k486270d/f110.image (and the following pages) and here http://gallica.bnf.fr/ark:/12148/bpt6k486270d/f138.image ...
3
votes
1answer
113 views

What happened to the Mertens constant in the strong prime twins conjecture ??

To estimate the amount of primes in an interval $\left(2,x\right)$ one might naively sieve by computing $ x \left(1-\dfrac{1}{2}\right)\left(1-\dfrac{1}{3}\right)...\left(1-\dfrac{1}{p_i}\right)$ ...
3
votes
1answer
170 views

Is this the way to estimate the amount of lucky twins?

To estimate the amount of prime twins between $3$ and $x$ we just take $x \prod_{p}(1-2/p)$ where $p$ runs over the primes between $3$ and $\sqrt x$. Lucky numbers are similar to prime numbers. Does ...
2
votes
1answer
206 views

How does sieve that Chen used to prove Chen's theorem work?

In the Number Theory for Computing, Song Y. Yan states that Chen used "complicated arguments based on sieve method", when proving what is now called Chen's theorem. How does this sieve work? Does it ...
2
votes
1answer
53 views

Quick way to iterate multiples of a prime N that are not multiples of primes X, Y, Z, …?

Is there a way to quickly iterate multiples of some prime $N$ while avoiding multiples of blacklisted primes $X$, $Y$, $Z$, ...? By quickly I mean is there a faster way than: Increment current ...
9
votes
2answers
198 views

“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). ...
3
votes
0answers
209 views

Question about recursive defined functions.

This question is about counting functions. With counting functions $F$ I mean functions from the positive integers to the positive integers that are strictly nondecreasing and can grow no faster than ...
0
votes
1answer
134 views

On the probability that $ 2x^2 + 1 $ is prime (quadratic residue)

I tried to compute the multiplicative inverse of the probability that $ 2 x^2 +1 $ is prime. (I'm aware that proving there are infinitely many such primes is not done yet, but let's ignore that for ...
0
votes
0answers
55 views

sum over prime index done by a weird sieve?

As you might have noticed i considered in 2 previous questions sums of the form $f(p_i x)$ where the sum is over the primes $p_i$ ( between some integer bounds : $a \leqslant p_i \leqslant b$ ) , $x$ ...
2
votes
0answers
153 views

How many co-primes are there for a big integer N over a specified interval?

How many co-primes are there for a big integer $N$ over a specified interval ? bounds of $N$ are $[1,10^9]$ and the interval is $[a,b]$ where ($1\leq a\leq b \leq 10 ^{15}$) and there are $100$ ...
1
vote
1answer
184 views

Numbering primes within a range.

$$n\ln n + n\ln\ln n−n < p_n < n\ln n+n\ln\ln n \mbox{ for } n\geq 6$$ This is the range where the $n$-th prime must lie. However sieving within this range generates a large number of primes. ...
1
vote
1answer
201 views

How to select the values X and Y in the Sieve Of Atkin Algorithm

I came to know Sieve of Atkin is the fastest algorithm to calculate prime numbers till the given integer. I am able to understand the sieve of Eratosthenes from wikipedia page but i am not able to ...
1
vote
4answers
1k views

Is it a bad idea to use a Sieve of Eratosthenes to find all primes up to very large N?

I need to write a program in C++ that finds all primes up to 2^32. I used a Sieve of Eratosthenes with multiple threads, but it only worked well up to about 10 million. After that it just takes too ...
4
votes
2answers
670 views

Erdős and the limiting ratio of consecutive prime numbers

The following is a piece of math lore from the late forties, which was described in an Intelligencer article entitled "The Elementary Proof of the Prime Number Theorem". It reads: Turán, who was ...
1
vote
1answer
332 views

Quadratic forms and prime numbers in the sieve of Atkin

I'm studying the theorems used in the paper which explains how the sieve of Atkin works, but I cannot understand a point. For example, in the paper linked above, theorem 6.2 on page 1028 says that if ...
4
votes
1answer
2k views

Understanding the Sieve of Atkin

I'm attempting to construct a program (in C++) that will count the prime factors of a given number for a Project Euler problem using the Sieve of Atkin, but I'm having trouble understanding a few ...
7
votes
1answer
239 views

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 ...
2
votes
4answers
521 views

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
votes
2answers
112 views

Estimating number of crossings for Erastothenes' Sieve

In this paper (2.1) I need to understand the formula for the total number of operations: $$\sum_{i=1}^{\pi(\sqrt n)}\frac{n}{p_i} \approx n\ln \ln n + O(n)$$ On a sidenote, since we're only checking ...
7
votes
2answers
526 views

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 ...