Questions tagged [twin-primes]
For questions on prime twins.
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Integer Parameterization of degree 2 equation: $(6x+5)y - x - 2 = (6w + 7)z - w + 4$
Finding a complete integer parameterization of $(6x+5)y - x - 2 = (6w + 7)z - w + 4$ has proved challenging. Can anyone lead me in the right direction? This is related to some nonprofessional research ...
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Brun's theorem and the twin prime conjecture
According to the following extract taken from Wikipedia, almost all prime numbers are isolated given Brun's theorem. Doesn't that mean that there is only a finite number of twin prime numbers (they ...
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If a pseudonorm function $N$ is continuous in a given topology, does the pseudometric topology formed by $d(x,y)=N(x-y)$ coincide with first topology?
Define $p_n\#= p_n p_{n-1} \cdots p_1$ for $n = 0$ to be $1$, then we have a function:
$$
N : \Bbb{Z} \to \Bbb{Z}, \\
N(x) = \left |-1/2 + \sum_{d\ \mid\ p_n\#} (-1)^{\omega d} \sum_{r^2 = 1 \mod d} \...
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What is a prime sieve method, and how did they help Zhang, Maynard and Tao?
At children's school we learned about the Sieve of Eratosthenes for sieving our primes from an interval of natural numbers.
I was surprised to hear that "sieve methods" were used to make ...
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Is it possible to have this overlap between Goldbach and the twin prime conjectures?
This question is related to this. But, here it is related Goldbach's conjecture.
Any even number greater than $4$ is the result of addition of two prime numbers one of which is the lower of a twin ...
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Is this conjecture about twin primes known to be false?
I'm not sure if this has been investigated before. This is a kind of strong twin prime conjecture
Define a first twin prime as the lower of a twin prime pair, while a second twin prime is the upper of ...
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$f(x) = \left\lfloor\frac{x- a}{b} \right\rfloor$ is continuous in Furstenberg's coset topology, so twin primes are counted by a continuous function.
Consider the function $f(x) = \left\lfloor\frac{ x - a}{b} \right\rfloor$ for fixed, $a, b\in \Bbb{Z}$, and $b \neq 0$.
Now consider the evenly-spaced integer topology (Also known as Furstenberg's ...
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How do two distinct, but possibly related, formulas each give rise to OEIS A067611?
BACKGROUND: The Sieve of Sundaram effectively identifies composite odd numbers, relying on the property that the odd numbers (i.e. numbers having no factors of $2$) are closed under multiplication. $...
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Distribution of primes in primitive Pythagorean triples
My Observation:
I've observed a pattern where for every pair of twin primes ($p$, $p+2$), there appears to be at least one primitive Pythagorean triple ($a$, $b$, $c$) such that one of the twin primes ...
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How many twin prime pairs between $6n$ and $36n^2$?
Is this a valid method to calculate a lower bound on the number of twin prime pairs that occur over $(6n$, $36n^2]$?
Divide the number line into groups of six, each of which contains a potential twin ...
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question on estimator for $\frac{\pi(n)}{n}$ and $\frac{\pi_2(n)}{\pi(n)}$
$\pi(n)$ and $\pi_2(n)$ represent the count of primes and count of twin primes $\leq n$ respectively.
Suppose we want to estimate $\frac{\pi(n)}{n}$. One way which obviously is not error-free is to ...
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broader meaning of twin prime constant?
It appears that the twin prime constant has meaning outside of the strict twin prime constant. I attempted to keep this post as short as possible.
Definitions:
Let $p,q$ represent primes and let $n$ ...
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Prove that at least two are the same $p = b^c + a , q = a^b + c , r = c^a + b$
Given a, b, c ∈ N p = bc + a ,
q = ab + c , r = ca + b we know that p q r are primes. Prove that at least two of the p ,q ,r are the same.
Edit:
i have tried with contradiction method.I assumed all ...
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Prime twin counting by $\pi_2(t^2) =^? \sum_{2<j<t^2} (-2)^{\omega(j)} (1/2)(\lfloor{\frac{t^2}{j}}\rfloor +\lfloor{\frac{t^2-2}{j}}\rfloor) +C$?
Let $\omega(n)$ count the number of distinct prime factors of the integer $ n \geq 2$. This $\omega(n)$ is called the prime omega function.
Inspired by these ideas :
Improved sieve for primes and ...
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About prime gaps ; $\sum_{m = 2}^{\ln_2^{*}(n)} \pi_{m}(n) \leq \pi(n)$?
Let $\pi(n)$ be the number of primes between $1$ and $n$.
Let $\pi_2(n)$ be the number of prime twins (gap $2$) between $1$ and $n$.
Let $\pi_3(n)$ be the number of prime cousins (gap $4$) between $1$ ...
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