Questions tagged [twin-primes]
For questions on prime twins.
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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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An argument for the infinitude of twin primes
Beyond 3, potential prime numbers occur at values equal to $6n\pm1$ where $n=1,2,3...\infty$. When both of these values are prime for a given $n$, a twin prime pair occurs.
Dividing the number line ...
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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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question on an extension of a theorem of Mertens
Mertens' theorem 3 states:
$$\lim_{n\to\infty}\prod_{p\le n}\left(1-\frac{1}{p}\right)=\frac{e^{-\gamma}}{\log n}$$
Separately, the twin prime constant $C_2$ is defined as:
\begin{align}
C_2 &= \...
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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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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 $A$ ...
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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 ...
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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:
$$ \lim\limits_{n\to \infty} \frac{\pi^2(n)}{n\pi_2(n)} = 0.7550363087870907 \cdots\cdots (1) $$...
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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 ...
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$n+1$ 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 $n \phi (n) + 1$ ...
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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 ...
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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 ...
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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{ ...
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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 ...
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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,\...
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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 ...
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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 ...
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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) ...
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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 $$f(x) = \sum_{d \mid p_{\...
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Number of divisors of composite number between and adjacent to twin primes
I am investigating properties of the number of divisors of composites between and adjacent to twin primes. When running some numeric calculations in Python (which are hopefully correct) I get the ...
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The series $\sum \frac{1}{p_iq_i}$ where $p_i,q_i$ are twin primes
I am interested in the series comprising the inverses of the products of twin prime pairs: $$\sum_i \frac{1}{p_iq_i}$$ where $p_i=6i-1,\ q_i=6i+1;\ (p_i,q_i) \in \mathbb P$. This series is equivalent ...
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For any $I = [p_{n} + 2, p_{n+1}^2 - 2]$ there exists an affine function in the first quadrant that lower bounds the twin prime average counter on $I$
Motivation
Lemma. $h_n(x) = \sum\limits_{d \mid p_n\#}(-1)^{\omega(d)} \sum\limits_{r^2 = 1 \pmod d} \left\lfloor \frac{x - r}{d}\right\rfloor$ counts the number of twin prime averages in the ...
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Every twin prime average $x \gt 6$ is the sum of two twin prime averages (Code checked up to $x \leq 1,000,000$).
If $p,q$ are a pair of twin primes, then $x = \dfrac{p+ q}{2} = q-1 = p+1$ is their twin prime average.
Conjecture. Every twin prime average $x \gt 6$ is the sum of two smaller twin prime averages, $...
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Is $\ 2\ 377\ 271\ $ the smallest number giving the desired twin prime pair?
I searched a twin prime of the form $$k\cdot 2023!\pm1$$ with positive integer $k$ as a project related to the current year. User hardmath claimed to have checked the range upto $k=2\ 200\ 000$ with ...
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The number of $n$ and $n+2$ that have at most seven prime factors
In Murty and Cojocaru, we have the theorem: As $x\rightarrow\infty$,
$$\#\{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 ...
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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 ...
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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\...
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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 (...
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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\...
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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.
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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 ...
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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 ...
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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
$$
Then I proved with the help of quadratic reciprocity, that ...
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Does this alternative way of calculating Twin Primes help to prove that there are an infinite number of Twin Primes?
I recently saw this video (https://www.youtube.com/watch?v=n4gmYjyI3vo) which explained a proof showing that all twin primes, when multiplied together, have a product where the digits of the product ...
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Is this twin prime-related sequence known?
For a positive integer $\ m\ $ define $\ k_m\ $ to be the smallest positive integer such that $$k_m\cdot m!\pm 1$$ form a twin-prime pair.
Has the sequence $(k_m)$ already been verified ?
The first $...
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A question on (trigonometric) prime counting function and twin prime counting function
Consider the following sum:
$$S(t)=\sum_{n=5}^t\sin^2\left(\frac{π\Gamma(n)}{2n}\right)$$
As we can see this approximates $π(t)$ i.e. prime counting function pretty well.
For details visit this paper ...
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Heuristic on the percentage of "lesser" twin primes congruent to 1 modulo 4
Refer to the smaller prime in a twin prime pair as a lesser twin prime. As an odd number, a lesser twin prime is congruent to either 1 or 3 mod 4: is anyone aware of existing heuristics which predict ...
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What is m/n as n tends to infinity?
Here m is in twin prime pair:(6m-1, 6m+1) and n is nth twin prime-pair.
I am just interested to know lower bound of difference of consecutive first twin primes as n tends to infinity.
For example in (...
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The strong twin conjecture can be transformed into the unsolvability of a particular Diophantine equation
Let us consider the strong twin conjecture:
For all positive integer $n$ there exist a prime $p$ such that
$$n+4<p<2^n2^4$$ and $p$ is a prime and $p+2$ is a prime
Since the inequalities and the ...
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A formula that counts exactly the twin prime averages occuring in an interval $[a,b]$ is surprisingly succinct!
Let $p_n$ denote the $n$th prime number. Let $p_n \lt a \lt b \lt p_{n+1}^2$ be any such integers. Their oddness or divisibility does not matter as in my previous posts, which makes this formula ...
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Major errors in an inclusion-exclusion application to counting twin primes in a certain interval.
Let $a,b,n,q$ all be odd numbers, and let $p_N$ be the $N$th odd prime, throughout.
Formula for the number $H(a,b,n) = \# \{a \lt x \lt b : x^2 - 1 = 0 \pmod n, x \text{ even}\}$. This is true since ...
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Can the prime counting function possibly satisfy this functional equation?
Let $a,b,n$ be odd natural numbers, $a \gt b$, and $n \gt 1$
Here's a proof by BillyJoe that the function,
$$h(a,b,n) := 2(\lfloor \dfrac{b}{n} \rfloor - \lfloor\dfrac{b}{2n} \rfloor - \lfloor \dfrac{...
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$H(n)=\lfloor\dfrac{b}{n}\rfloor- \lfloor \dfrac{a}{n} \rfloor=$ (roughly) # odd pairs $o, o+2 \in [a,b]$ such that $n \mid o$ or $n \mid o+2$
I came up with the following formula and deleted that question so that I don't have two questions on the same formula.
Conjecture. Let $a, b, n \in 2\Bbb{N} + 1$ be odd natural numbers. Then the ...
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Is there a formula for $|H_n|$, where $H_n = \{ $ units $u \pmod n$ such that $u^n = u, \}$ is the group of $(n-1)$th roots of unity modulo $n$?
Denote the group of solutions $X$ modulo $n$ to
$$
X^{m} = X \pmod n
$$
by $H(m,n)$. Then $H(m,n)$ is a subgroup of $G_n = \Bbb{Z}_{n}^{\times}$ the group of units modulo $n$. Note that $H(n-1,n) = ...
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Potential infinite, fast growing subsequence of twin primes
Probably the most interesting part of this discussion is about twin primes of the form $6x\pm 1$, with $x=4\cdot(5\cdot 7\cdot 11\cdot 13\cdot 17
\cdot 19\cdot 23)$ being a typical example, and the ...
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Observation about twin primes: is it true? If so, why?
I noticed today that every set of twin primes except for $(3,5)$ and $(5,7)$ seems to have one of the two primes that can be represented by the sum of two squares. For example:
\begin{eqnarray*}
13=3^...
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For twin primes $p$ and $q$, prove there is an integer $a$ such that $p|(a^2-q)$ if and only if there is an integer $b$ such that $q|(b^2-p)$.
For twin primes $p$ and $q$, prove there is an integer $a$ such that $p|(a^2-q)$ if and only if there is an integer $b$ such that $q|(b^2-p)$.
Algebraic substitution using $p=q+2$ and the definition ...
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Full set of congruences uniquely defining all twin primes
I am looking at the sequence A002822 which essentially is equivalent to the twin prime sequence. Based on initial investigations, it has the following peculiar congruential structure.
Pattern A
For ...
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