For questions on prime twins.

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Twin Primes between $n$ and $2n$

Is it theoretically possible for there to always be a twin prime pair between $n$ and $2n$ for all sufficiently large $n$ (assuming of course that there are infinitely many twin primes) or would this ...
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3answers
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What is wrong with this proposed proof of the twin prime conjecture?

I was thinking on the twin prime conjecture, that there are an infinite number of twin primes... I came up with a proof. I have to think that it is incomplete or wrong, because many great minds ...
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2answers
413 views

Modified Euler's Totient function for counting constellations in reduced residue systems

I am working on a modified totient function for counting constellations in reduced residue systems for the same range that Euler's totient function is defined over. This post is separated into three ...
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1answer
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Infinite “Twins” in reduced residue systems modulo primorials

The Lth primorial ($p_L\#$) is the product of the first L prime numbers. The reduced residue system modulo $p_L\#$ is any set of positive integers with cardinality equal to the totient of $p_L\#$ ...
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0answers
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Is there is any result claiming that there cannot be any other twin Mersenne primes?

There are 3 known Twin Mersenne Primes: $M3$ and $M5$, $M5$ and $M7$, $M17$ and $M19$. More precisely, if both $M(p)$ and $M(p+2)$ are both prime, then they are called Twin Mersenne Primes. My ...
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5answers
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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 ...
7
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1answer
195 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 ...
2
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0answers
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Test: Total number of twin primes in the vicinity of twin primes: how can I calculate the upper and lower bounds of the results?

I have performed the following test, and according to the results, I do not know how to define a function to calculate the limit of the lower and upper bounds of the data results. Besides, looking at ...
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Let $g_n^k=p_{n+k}-p_n$, where $p_n$ is the $n$th prime. Does there exist $g_{k+1}^1=2$ such that $g_1^k,g_2^k,\ldots$ is a “Gilbreath sequence?”

Call $(S_i)_{i=1}^{\infty}$ a Gilbreath sequence if $1=\lvert S_2-S_1\rvert=\lvert \lvert S_3-S_2\rvert-\lvert S_2-S_1\rvert\rvert=\cdots$, i.e., if the sequence can be substituted for the primes in ...
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1answer
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Asymptotic Expression for the Twin Prime Counting Function

A variation on a previous question I asked, which has garnered no responses. I'll attempt to be more lucid: Let $\pi_2(x)$ be the twin prime counting function and $\pi(x)$ be the prime counting ...
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15answers
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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 ...
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0answers
64 views

Prime Reflections

How would you describe the following pattern?: For each primorial from 30 onward, there exists a pattern in the arrangement of the prime factors of the composite numbers which I call "the mirror ...
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2answers
149 views

What if a conjecture were provably unprovable?

Suppose we found a proof that "The Twin Prime Conjecture cannot be proven", without any conclusion as to the conjecture itself being true or false. Is it then possible for the conjecture to be true? ...
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1answer
63 views

Can we abstractly construct this ring and what is it isomorphic to?

Define $R = \Bbb{Z}[X_1, X_2, \dots]$. Then place on $R$ the relations $$ X_1 + 1 = X_2, \\ X_2 + 2 = X_3, \\ X_3 + 2 = X_4, \\ X_5 + 2 = X_6, \dots \\ X_{2k-1} + 2 = X_{2k}, \ \forall k \geq 3 ...
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2answers
182 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 ...
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0answers
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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 ...
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0answers
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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 ...
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2answers
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Conjecture on twin primes

Let $p$ and $p+2$ be both prime. I conjectured (with my ignorance) that $$p^{\frac{p+1}{2}}\equiv -1\mod{(p+2)}$$ except for $p=17,41,71,137, 191, 239....$ I verified this on Mathematica. So for ...
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1answer
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Is there a generalization of the twin prime conjecture to rings or certain rings?

The question's in the title. For instance, if $R$ contains $2$ then there are an infinite number of pairs of prime principal ideals $(p),(q)$ such that $p = q + 2$. I just made that up and it's ...
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3answers
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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 ...
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3answers
356 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, ...
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0answers
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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
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1answer
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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 ...
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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
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1answer
251 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 ...
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3answers
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Bounds on the Twin Prime counting function

Maybe I'm mistaken, but wouldn't it suffice to show that the twin prime counting function has any kind of strictly increasing lower bound to show that there are infinitely many twin primes?
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1answer
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Can we determine which statements are incomplete due to Godel?

Due to Godel's incompleteness theorems we know that there are true statements in a system that cannot be proven with that system. My questions are 1) can we tell which statements in a system are the ...
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0answers
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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, \ ...
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0answers
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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 ...
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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 ...
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1answer
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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 ...
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2answers
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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 ...
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2answers
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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 ...
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2answers
95 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 ?
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1answer
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Why is factorization of large number hard

Why factoring a number is difficult compared to finding out if it is prime (which can be done in polynomial time) ? I would think they might be of similar difficulty in terms of computational ...
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1answer
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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 $ ...
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1answer
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Idea about Twin-Primes and the generation of natural numbers

Years ago (6 years to be exact) I was fascinate by prime-twins, and still I am, but the years went by and I almost forgot about it until yesterday. I found my notes again and I don't know if I am on ...
3
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1answer
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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 ...
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1answer
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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)$ ...
2
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1answer
263 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 ...
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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?
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3answers
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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 ...
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2answers
141 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 ...
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2answers
156 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 = ...
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1answer
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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 in´Čünitely many primes $p$. b) Twin primes are those difference by 2. Show that 5 is the ...
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1answer
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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 ...
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1answer
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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 ...