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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 ...
ServingSpy's user avatar
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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 ...
David's user avatar
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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} \...
SeekingAMathGeekGirlfriend's user avatar
4 votes
1 answer
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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 ...
Penelope's user avatar
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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 ...
Zuhair's user avatar
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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 ...
Zuhair's user avatar
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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 ...
SeekingAMathGeekGirlfriend's user avatar
1 vote
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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. $...
Keith Backman's user avatar
2 votes
2 answers
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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 ...
Nicholas Joseph's user avatar
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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 ...
Ricky Vesel's user avatar
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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 ...
sku's user avatar
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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 $n$ ...
sku's user avatar
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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 ...
john's user avatar
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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 ...
mick's user avatar
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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 $1$ ...
mick's user avatar
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Is there any twin prime representing function?

There are many prime representing functions. For example, $\lfloor A^{3^n} \rfloor$ is prime representing function because for all positive integers n ,it generates a different prime number. Here $A$ ...
Severus' Constant's user avatar
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If $n$ is a Poulet number then $n+2$ is not a Poulet number?

Consider Fermat pseudoprimes to base $2$, also called Sarrus numbers or Poulet numbers. Inspired by prime twins it makes sense to consider : Conjecture : If $n$ is a Poulet number then $n+2$ is not a ...
mick's user avatar
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10 votes
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An interesting finding on twin primes

I was doing some research on prime gaps including twin primes and this led me to this finding, which is: $$ \lim\limits_{n\to \infty} \frac{\pi^2(n)}{n\pi_2(n)} = 0.7550363087870907 \cdots\cdots (1) $$...
sku's user avatar
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All twin prime averages in the range $[9, 119]$ are of the form $6(5[3(z-x)]_{\pmod 7} + x)$ for some $x \in \{0,2,3\}, z \in \{0,2,3,4,5\}$.

Question. Can we come up with a general formula $f(x_5, x_7, x_{11}, \dots, x_{p_n})$ such that each twin prime average $a \in [p_n + 2, p_{n+1}^2 - 2]$ is expressible as $f(x_5, \dots, x_{p_n})$ for ...
SeekingAMathGeekGirlfriend's user avatar
16 votes
2 answers
484 views

$n+1$ and $n \phi (n) + 1$ are both perfect squares if and only if $n$ is a product of twin primes?

I'm trying to prove the following conjecture concerning twin primes and Euler's totient function, which I have verified for $n$ up to 1 billion. For all $n \in \mathbb{N}$, $n+1$ and $n \phi (n) + 1$ ...
JMP's user avatar
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$A(x) = \prod_{p,q \in \Bbb{P}} (1 - \frac{x^2 - 1}{pq}) = 0$ if and only if $x$ is a twin prime average?

Conjecture. For any integer $x \geq 1$, we have: $$ A(x) = \prod_{p,q \in \Bbb{P} \\ p \lt q} \left(1 - \frac{x^2 - 1}{pq}\right) = 0 $$ if and only if $x$ is a twin prime average. How can we prove ...
SeekingAMathGeekGirlfriend's user avatar
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1 answer
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New method to generate twin primes.

Is this a valid method to generate twin primes? Let $q = \left\lfloor \sqrt{n} \right\rfloor$. If $n = q \cdot (q + 2)$ and $\quad \gcd(n,m) = 1 \quad \forall m \in \left[ \left\lfloor \frac{n}{2} \...
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Potential progress concerning the twin prime conjecture

The twin prime conjecture posits that there are an infinite number of twin primes, or equivalently that there is no largest twin prime pair. I conjecture more specifically that for any twin prime pair ...
Keith Backman's user avatar
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Can we write down a formula for the simultaneous images of the $C_q$?

Fix $n \in \Bbb{N}$. Define $C_2(x) = 2x, \ C_3(x) = 3x$, $C_5(x) = 2 [\frac{x + 2}{3}] + [\frac{ x + 1}{3}] + 2 [ \frac{x}{3}]$, more generally for prime $p_n \geq q \geq 5$ define $C_q(x) = 2[\frac{ ...
SeekingAMathGeekGirlfriend's user avatar
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A corollary of Brun's pure sieve from Opera de Cribro by Friedlander and Iwaniec

I'm stuck on the one (if not many) step of the proof of Corollary 6.2 (page 58) from the book "Opera de Cribro" by Friedlander and Iwaniec. The statement is a corrollary of Brun's pure sieve ...
LiangPrime's user avatar
2 votes
2 answers
185 views

Can the sum of squares of odd primes equal the square of an odd prime?

In this question it was shown that collections of $8k+1$ odd squares can be found that sum to an odd square. In his answer, Denis Shatrov provided an algorithm by which $8k$ odd numbers $\{a_1,a_2,\...
Keith Backman's user avatar
1 vote
0 answers
72 views

primes of the form 2+pq

Is it always possible to demonstrate the existence of at least one prime number of the form 2 + pq, where p is an arbitrarily large prime number and q is a prime number greater than p? Other word, if ...
conjectures's user avatar
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1 answer
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Is some twin prime average the sum of two twin prime averages, two ways?

Accoring to this question and a linked duplicate, it's been verified empirically up to some number that all twin prime averages greater than six, are the sum of two smaller twin prime averages. I was ...
it's a hire car baby's user avatar
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Describing the probability that n is of the form $K^2+1$ , and it is a prime number.

I was reading a book called "Math Talks for Undergraduates" by Serge Lang. I was introduced to the Prime Number Theorem that states that $pi(x)$ (representing the number of primes $<=$ x) ...
Epsilon's user avatar
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Conjecture: It is not true that $2$ eventually always divides $f(x) = \sum_{d \mid p_{\sqrt{x+1}}\#} (d \mid x^2 - 1)$.

Lemma. Let $(d \mid \cdot ) : \Bbb{Z} \to \{0,1\}$ be whether $(1)$ or not $(0)$ $\ d$ divides the input. There exists no $N \in \Bbb{N}$ such that $\forall x \geq N$ we have $$f(x) = \sum_{d \mid p_{\...
SeekingAMathGeekGirlfriend's user avatar
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Number of divisors of composite number between and adjacent to twin primes

I am investigating properties of the number of divisors of composites between and adjacent to twin primes. When running some numeric calculations in Python (which are hopefully correct) I get the ...
Victor Galeano's user avatar
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The series $\sum \frac{1}{p_iq_i}$ where $p_i,q_i$ are twin primes

I am interested in the series comprising the inverses of the products of twin prime pairs: $$\sum_i \frac{1}{p_iq_i}$$ where $p_i=6i-1,\ q_i=6i+1;\ (p_i,q_i) \in \mathbb P$. This series is equivalent ...
Keith Backman's user avatar
4 votes
0 answers
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For any $I = [p_{n} + 2, p_{n+1}^2 - 2]$ there exists an affine function in the first quadrant that lower bounds the twin prime average counter on $I$

Motivation Lemma. $h_n(x) = \sum\limits_{d \mid p_n\#}(-1)^{\omega(d)} \sum\limits_{r^2 = 1 \pmod d} \left\lfloor \frac{x - r}{d}\right\rfloor$ counts the number of twin prime averages in the ...
SeekingAMathGeekGirlfriend's user avatar
3 votes
0 answers
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Every twin prime average $x \gt 6$ is the sum of two twin prime averages (Code checked up to $x \leq 1,000,000$).

If $p,q$ are a pair of twin primes, then $x = \dfrac{p+ q}{2} = q-1 = p+1$ is their twin prime average. Conjecture. Every twin prime average $x \gt 6$ is the sum of two smaller twin prime averages, $...
SeekingAMathGeekGirlfriend's user avatar
1 vote
1 answer
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Conjecture. If $n \in \Bbb{N}\setminus 1$ is not a twin prime average, then $n^2 - 1$ is not square-free.

From the evidence: ...
SeekingAMathGeekGirlfriend's user avatar
5 votes
2 answers
170 views

Is $\ 2\ 377\ 271\ $ the smallest number giving the desired twin prime pair?

I searched a twin prime of the form $$k\cdot 2023!\pm1$$ with positive integer $k$ as a project related to the current year. User hardmath claimed to have checked the range upto $k=2\ 200\ 000$ with ...
Peter's user avatar
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1 vote
1 answer
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The number of $n$ and $n+2$ that have at most seven prime factors

In Murty and Cojocaru, we have the theorem: As $x\rightarrow\infty$, $$\#\{n\le x: n \text{ and } n+2 \text{ have at most seven prime factors}\}\gg \frac{x}{(\log x)^2}$$ The proof uses Brun's sieve ...
James2390's user avatar
2 votes
1 answer
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Consequence of Brun's Sieve

I'm following a text that uses Brun's sieve (as described in https://pages.cs.wisc.edu/~cdx/Sieve.pdf in Theorem 2.2.2) to prove the following theorem: For any $\alpha\in\mathbb{Z},\alpha\neq0$, we ...
James2390's user avatar
2 votes
0 answers
143 views

Questions Related to the Twin-Prime Conjecture

For the sieve of Eratosthenes, let $E_k$ be the number of elements left after removing all primes up to $p_k$ and their multiples from the set $\left\{1,2,3,...p_k\#\right\}$ where $p_k\#=\prod\...
Steven Clark's user avatar
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Twin primes and circles.

For $n$ prime, noticed a pattern whenever a twin prime $(x,y) \in \mathbb{Z}^{+}$ is the only point satisfying the circle $x^2+y^2=n^2 + (n + 2)^2$ then $\Large \frac{x^2+y^2}{2}$ is prime. Example (...
vengy's user avatar
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A property of twin primes

$(a,b)$ is a couple of twin primes such that $b=a+2$ and $a > 29$. Let $N = 4^b$ and $q$ the quotient which results from the division of $N$ by $a$ and $r$ is the remainder. We calculate $P = (q\...
Craw Craw's user avatar
1 vote
0 answers
149 views

Is there integer $n>3$ such that both $n!-1$ and $n!+1$ are prime?

Is there integer $n>3$ such that both $n!-1$ and $n!+1$ are prime? For $n=3$ we know that $5=3!-1$ and $7=3!+1$ are prime.
Mohammad Abry's user avatar
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0 answers
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Full derivation inside of twin prime statement in terms of multiplicative arithmetic functions. How can the last formula be rearranged?

Let $(\cdot\mid\cdot) : \Bbb{N}\times\Bbb{N} \to \Bbb{Z}_2$ be the divisibility function which takes on the value $(x|y) = 1$ whenever $x$ divides $y$ and the value $(x|y) = 0$ whenever it does not ...
SeekingAMathGeekGirlfriend's user avatar
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0 answers
163 views

Assume that the Twin prime conjecture is true, prove that there are infinitely many pairs of positive integers m and n such that $\phi(m)=\sigma(n)$

From a comment, I have corrected my proof. Here's what I have now. The Twin Prime Conjecture sates: There are infinitely many prime numbers $p$ for which $p+2$ is also a prime number.We consider 61 ...
Jack Hilton-Jones's user avatar
1 vote
0 answers
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Twin Primes of the form $3k-1, 3k+1$

I wanted to discuss something. Yesterday I thought about the twin prime conjecture and I constructed numbers of the form $$ 3k-1, 3k, 3k+1 $$ Then I proved with the help of quadratic reciprocity, that ...
calculatormathematical's user avatar
-3 votes
1 answer
118 views

Does this alternative way of calculating Twin Primes help to prove that there are an infinite number of Twin Primes?

I recently saw this video (https://www.youtube.com/watch?v=n4gmYjyI3vo) which explained a proof showing that all twin primes, when multiplied together, have a product where the digits of the product ...
zoplonix's user avatar
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Is this twin prime-related sequence known?

For a positive integer $\ m\ $ define $\ k_m\ $ to be the smallest positive integer such that $$k_m\cdot m!\pm 1$$ form a twin-prime pair. Has the sequence $(k_m)$ already been verified ? The first $...
Peter's user avatar
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4 votes
1 answer
140 views

A question on (trigonometric) prime counting function and twin prime counting function

Consider the following sum: $$S(t)=\sum_{n=5}^t\sin^2\left(\frac{π\Gamma(n)}{2n}\right)$$ As we can see this approximates $π(t)$ i.e. prime counting function pretty well. For details visit this paper ...
TPC's user avatar
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2 votes
0 answers
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Heuristic on the percentage of "lesser" twin primes congruent to 1 modulo 4

Refer to the smaller prime in a twin prime pair as a lesser twin prime. As an odd number, a lesser twin prime is congruent to either 1 or 3 mod 4: is anyone aware of existing heuristics which predict ...
xion3582's user avatar
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1 vote
1 answer
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What is m/n as n tends to infinity?

Here m is in twin prime pair:(6m-1, 6m+1) and n is nth twin prime-pair. I am just interested to know lower bound of difference of consecutive first twin primes as n tends to infinity. For example in (...
Devansh Singh's user avatar

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