5
votes
1answer
106 views

Zhang's theorem and Polignac's conjecture

Yitang Zhang made a groundbreaking discovery when he proved that there are infinitely many pairs of prime numbers which differ by less than $70,000,000$. Zhang's theorem has been significantly ...
1
vote
2answers
168 views

About twin primes and their happy mothers.

Let's say that a positive integer $n$ is a happy mother if $6$ divides $n$ and $(n-1,n+1)$ is a pair of twin primes. Is the difference between two consecutive happy mothers necessarily a happy mother ...
1
vote
0answers
44 views

About a paper by Gold & Tucker (characterizing twin primes)

I've carefully looked at the questions on prime and twin prime, but the following question seems not to habe been asked before. Context: In the paper by Jeffrey F. Gold and Don H. Tucker titled A ...
1
vote
0answers
32 views

Estimating the Twin prime constant

http://numbers.computation.free.fr/Constants/Primes/twin.html it says: "This last constant occurs in some asymptotic estimations involving primes and it's interesting to observe that it may be ...
0
votes
0answers
37 views

Lower bound for twin prime counting function

We know an upper bound for twin prime counting function. If i'm not mistaken it was (x*loglogx)/(logx*logx). Do we have a lower bound for it? Is it the same function? I mean is there c constant such ...
3
votes
1answer
80 views

About Brun's Theorem

http://arxiv.org/pdf/1401.7555.pdf On the page 8 there is a proof of Brun's theorem. $$\large-\int_1^{\infty}\pi_2(x)\;\mathrm{d}\left(\frac1{\lfloor x \rfloor}\right)=-\sum_{n\ge ...
0
votes
2answers
166 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...)
4
votes
1answer
159 views

Can the twin prime conjecture be solved in this way?

After some research, I have discovered that proving the statement; There exist an infinite number of positive integers K such that; $K \neq 6ab \pm a \pm b$ and $K \neq 6ab \mp a \pm b$ is ...
-7
votes
1answer
475 views

Twin prime conjecture - please check proof. [closed]

Prove infinite twin primes using deduction: L M R T(n) 11 12 13 1 17 18 19 2 23 24 25 3 (3+5n) 29 30 31 4 35 36 37 5 (5n, 5+7n) 41 42 ...
70
votes
14answers
7k views

Infiniteness of non-twin primes.

Well, we all know the twin prime conjecture. There are infinitely many primes $p$, such that $p+2$ is also prime. Well, I actually got asked in a discrete mathematics course, to prove that there are ...
1
vote
0answers
72 views

Is this statement equivalent to Goldbach's conjecture

Given a number $n\ge 3$, then one of these is true: \begin{equation} \begin{cases}2n = (6m-1)+P, \ \ \ P \in \mathbb P, \ 6m-1 \in \mathbb P, \ 6m+1 \in \mathbb P \ \ \ \ (1) \\ 2n-1 \in \mathbb P, \ ...
1
vote
0answers
71 views

Estimating total number of twin primes

Taking my notation from a previous question Define a function $P_6$ as $$P_6(n)=\begin{cases} 0, \ \ 6n-1 \not\in \mathbb P \wedge 6n+1 \not\in \mathbb P \\ 1, \ \ (6n-1 \not\in \mathbb P \wedge ...
6
votes
0answers
89 views

For primes $P_1$ and $P_2$, exists a prime $P_3$ that both $P_i + 6P_3$ is a prime

I was thinking about twin primes and I came to ask this question: If we have two distinct primes $P_1$ and $P_2$ which are both greater than $3$, then does there always exist a prime $P_3$ such that ...
4
votes
2answers
81 views

Twin prime “test” via congruence

I decided to try getting a test for a "twinness" of a prime via Wilson's theorem. Wilson's theorem says that integer $n > 1$ is a prime iff $$(n-1)! \ \equiv -1 \pmod n $$ Now, if both $n$ and ...
1
vote
1answer
82 views

Let $p_k$ be the $k$th prime, can it be shown for $p \ge 5$, that there is not always a twin prime between $p_k^2$ and $p_{k+1}^2$?

For any primorial $p_k \ge 3$, $p_k\#$, there are $$\prod_{2\le{i}\le{k}} (p_i-2)$$ distinct instances of $x,x+2$ that are relatively prime to $p_k\#$. If any of these pairs are less than ...
2
votes
2answers
89 views

Assuming there exist infinite prime twins does $\prod_i (1+\frac{1}{p_i})$ diverge?

Assume there are an infinite amount of prime twins. Let $p_i$ be the smallest of the $i$ th prime twin. Does that imply that $\prod_i (1+\frac{1}{p_i})$ diverges ?
0
votes
1answer
53 views

Why does a non-zero density function not imply infinitude of what it measures?

Consider the following density function for the twin primes: Numbers $x-2$, $x-4$ are twin primes iff: $x \ne 2,4 \ mod \ 2 $ $x \ne 2,4 \ mod \ 3 $ $x \ne 2,4 \ mod \ 5 $ $x \ne 2,4 \ mod \ 7 $ ...
30
votes
2answers
6k views

Yitang Zhang: Prime Gaps

Has anybody read Yitang Zhang's paper on prime gaps? Wired reports "$70$ million" at most, but I was wondering if the number was actually more specific. *EDIT*$^1$: Are there any experts here who ...
6
votes
3answers
1k views

What would be the immediate implications of a formula for prime numbers?

What would be the immediate implications for Math (or sciences as a general) if someone developed a formula capable of generating every prime number progressively and perfectly, also able to prove (or ...
3
votes
1answer
123 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
179 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
226 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 ...
7
votes
3answers
199 views

If the set of primes where $p$, $p+2$ is infinite, would this imply that the set of $p$ and $p+2n$ is also infinite?

If the set of primes $p$ such that $p+2$ is also prime is infinite, would this imply that the set of primes such that $p+2n$ where $n$ is any positive integer for each pair is also infinite?
4
votes
4answers
483 views

Determining the next Twin Prime?

A really simple I question I guess. Is there an algorithm or method such that given an integer $N$ there is a way to determine the next twin prime pair greater than $N$? If yes, then could you please ...
2
votes
3answers
85 views

Whether twin primes satisfy this one?

It seems that difference of squares of any twin primes $+1$ will always lead to number which might be a) A square of a twin prime b) Itself a twin prime $C$ = ($A^2$-$B^2$ )+$1$ ------> $(1)$ Where ...
0
votes
2answers
135 views

Primes and Twine primes and their sums.

Need good discussion for $12|(p + p+2)$, where $p,p+2$ are primes and $> 3$. Why $12$ divides the sum of twin primes? $a, ar, ar^2, \ldots $ is a Geometric series. I would like to place $a = ...
0
votes
2answers
128 views

Twin primes and modulo

I am so exited to learn math from this site. I posted the question today and I got good replies from members today itself. I will try to answer other number Theory questions in near future. With same ...
4
votes
3answers
305 views

Twin primes satisfy the congruence?

I need a justification for my observation. In general, we can list twin prime pairs in $(6n-1, 6n+1)$, where $n$ is some positive number. Of course, this is valid except $(3, 5)$. Now, I construct, ...
2
votes
1answer
156 views

some problems related to primes

I would like to learn the following: a) Prove that the equation $1 + x + x^2 = py$ has integer solutions for infinitely many primes $p$. b) Twin primes are those difference by 2. Show that 5 is the ...
1
vote
1answer
63 views

What can I say about $x^4 \equiv -4 \mod p$ where $p$ is prime?

What can I say about $x^4 \equiv -4 \mod p$ where $p$ is prime? In general what can I do with powers that are greater than $2$ and where I cannot use reciprocity, legendre/jacobi etc... In general ...
2
votes
1answer
126 views

2 dimensional cellular automaton for prime twins?

Is there a 'simple' 2 dimensional cellular automaton to generate all prime twins ? With 'simple' I mean not too many states per cell and not so many rules. Thus a universal turing machine equivalent ...