Tagged Questions

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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 ...
170 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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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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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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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 ...
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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 ...
83 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 ...
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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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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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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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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 ...
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 ...
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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 $x \left(1-\dfrac{1}{2}\right)\left(1-\dfrac{1}{3}\right)...\left(1-\dfrac{1}{p_i}\right)$ ...
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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 ...
229 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 ...
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?
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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 ...
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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 ...