Questions tagged [prime-twins]

For questions on prime twins.

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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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Why does it appear that modern approaches to the Twin Prime Conjecture focus on optimization rather than construction? [closed]

I am wondering what makes it difficult to approach the twin prime conjecture by construction. I have only casual knowledge of the subject so far, so I apologize for any ignorance in advance. The ...
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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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The formula that counts the number of averages of $2k$-separated prime pairs in the interval $[n + 2k, (n+1)^2 - 2k]$ has the following form.

Let $k \geq 1$ and $n \geq k+1$. Then the formula: $$ f(k,n) := \sum_{d \mid n\#} (-1)^{\omega(d)}\sum_{c \mid d \\ \gcd(c, 2k) = 1} \left(\lfloor \dfrac{(n + 1)^2 - 2k - x_{c,d}}{d} \rfloor + \...
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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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Trick for an inclusion-exclusion of $\left|[a,b] \cap \left(\bigcup_{m = p_i}(m\Bbb{Z} -2)\cup (m\Bbb{Z})\right)\right|$?

Let $p_i$ denote the $i$th prime number, and if not specified otherwise, a variable takes its range of values in $\Bbb{Z}$. Define: $$ F(m,n) = \left\lfloor \dfrac{b - n}{m}\right\rfloor - \left\...
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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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Successive intervals bounded by numbers of the form $(6k+1)^2$ contain (approximately) equal numbers of twin primes

It has frequently been conjectured that intervals bounded by squares of one type or another will always contain one or more twin primes; see for example (1), (2), (3), (4), and (5). I present a ...
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6 votes
1 answer
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Is it possible that every $P$ as a prime number, can be expressed as a prime factor of $E$ such that $E$ is the sum of a pair of twin primes?

Curious about the Goldbach conjecture, and reading about twin primes, I was wondering if it is possible that every prime number as $P$, can be expressed as a prime factor of at least one $E$ such that ...
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What does it mean if the twin prime conjecture is true or false?

From what I understand, the main premise of the twin prime conjecture is "Are there an infinite number of twin primes?" And twin primes are prime numbers that are separated by two. Examples ...
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Alternating sum of the reciprocals of the twin primes $\sum(\frac1p-\frac1{p+2})$

I ask if the limit of the alternating sum of the reciprocals of the twin primes $$\sum_{p,\,p+2\,\in\,P}\Big(\frac1p-\frac1{p+2}\Big)=\frac13-\frac15+\frac15-\frac17+\frac1{11}-\frac1{13}\;+\;...=0....
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1 vote
1 answer
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Related to the sum of consecutive primes

Yesterday I saw this question: A question about divisibility of sum of two consecutive primes (you should read the OP to understand the full problem), it just asks to prove that for all $k\in \mathbb ...
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Why are some numbers "paired" in their prime distribution

Sorry I'm not sure exactly how to word the question. I was exploring off-by-one primes for each number, as I found it curious enormous primes were searched to be one off a power of $2$, and all primes ...
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Generating new prime number by adding $40$ to larger number of twin primes

A long time ago one of my classmates claimed he discovered a formula for prime numbers and he became so famous among students and our teacher. If we have two digits twin prime numbers (primes which ...
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A slight generalization of the Sieve of Sundaram that might shed light on the $6n \pm 1$ phenomenon of sequence A002822.

There's the $n$ such that $6n \pm 1$ is a twin prime pair sequence: https://oeis.org/A002822 It contains all twin prime averages (divided by $6$) other than $4$. Notice this sequence: Positive ...
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3 votes
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In an infinite consecutive set of only all the prime numbers, should we expect consecutive twin prime numbers to exist infinitely?

I am aware that in an infinite consecutive set of all positive integers, in theory there should be infinite twin prime numbers, but let's imagine an infinite set of only all the prime numbers. Here ...
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Do certain differences of two primes occur infinitely often?

This question concerns the generalization of certain characteristics of twin primes to a broader class of pairs of primes, and whether the generalized formulation might be used to provide insights ...
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3 answers
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A simple quadratic reformulation of the twin prime conjecture?

Due to https://oeis.org/A024702 we have that $p^2 - 1 ≡ 0$ (mod 24). For twin primes, we then must also have that in this case $(p+2)^2 - 1 ≡ 0$ (mod 24), which is the same as saying that $p^2 + 4p + ...
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9 votes
2 answers
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Can a prime number be bigger than the sum of adding the previous twin primes (other than 13)?

A simple heuristic of the first million primes shows that no prime number can be bigger than the sum of adding the previous twin primes. Massive update: @mathlove made a comment that leaves me ...
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2 votes
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Are those some new symmetries among the Twin Primes?

For each pair of Twin Primes $(p_a;p_b)$, the following operation gets applyed: Concatenation of the pair to one real number $R_{ab}$ by using a comma inbetween for separation. Calculating the square ...
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Are there any plausible arguments for the infinity of right shifted prime numbers? [closed]

By "right shifted prime numbers" I mean prime numbers of the form: $p_r \equiv$ $ 1 $ $mod $ $6$. $p_l \equiv$ $5$ $mod$ $6$ on the other side would be a left shifted prime number. Since all ...
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Slightly weaker than twin prime conjecture, using $\Omega: \Bbb{Q}^{\times} \to \Bbb{Z}$

Necessary background. Let $\Bbb{Q}^{\times} \xrightarrow{\Omega} \Bbb{Z}$ be the group hom given by $\Omega$ from the study of arithmetic functions (you can extend it uniquely (and homomorphically) to ...
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2 votes
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Correct reasoning for a finite number of twin primes next to the lcm of first N numbers?

Does the following argument make sense? Let ${\rm lcm}(1,...,N)$ be the least common multiple of the first $N$ natural numbers. In: Highest Twin primes such that the number in between twins is the $\...
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Highest Twin primes such that the number in between twins is the $\text{lcm}$ of first $N$ numbers

$\text{lcm}(1,2,3,4,5,6,7) = 420$ and this number is placed between two Twin primes: $419,\ 421$. This happens again for $\text{lcm}(1,2,3...19) = 232792560$ and $\text{lcm}(1,2,3,4,...,47)=...
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How can prime numbers be identified through characteristics of matrix/grid structures formed by polyhedra?

Have you come across a similar theory to predict prime numbers? I believe that I have identified a pattern by which prime numbers occur which is in the form: N = A(P,n) + B …where A is a function of a ...
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-3 votes
1 answer
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Why the twin prime conjecture isn't proved already by Euclid's theorem? [duplicate]

I was wondering how Euclid showed that there are infinitely many primes by generating a prime number from finitely many primes, and if it could be used to answer if there are infinitely many pairs of ...
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3 votes
1 answer
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Prime Gap $\lim\limits_{n \to \infty} \frac{3g_{n}^2}{p_{n}}=0$ from observation to proof

The following limit (after analyzing 3 successive primenumbers) was found: $$\lim_{n \rightarrow \infty}\frac{3g_{n}^2}{p_{n}}=0$$ $$g_{n}\ll \sqrt{\frac{p_{n}}{3}}$$ Thanks to comments I could trace ...
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1 answer
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Indicator equation $f(n) = n$ for $n \in \Bbb{N}$ such that $n \pm 1$ is a pair of twin primes.

Consider the formula: $$ f : \Bbb{N} \to \Bbb{C} \\ f(n) := 2 + \sum_{a = 1}^{n-2} \exp\left({\dfrac{2\pi i}{\gcd(a, n^2-1)}}\right) $$ If $n \pm 1$ is a pair of twin primes, then $f(n) = n$. This is ...
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4 votes
2 answers
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Euler products, Merten's theorems, and an unexpected result

I'm going to start by saying I'm mostly out of my depth here. I'm an amateur recreational mathematician. But I've been looking at the Twin Prime Conjecture lately, because it is so fascinating. Easy ...
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Why does this ratio $5$ occur relating prime twins and sophie germain primes?

Consider the twin primes. So we consider $3,5,7,11,13,17,19,29,31,...$ Call the $n$-th element in the sequence $t(n)$. Now consider the Sophie Germain primes ( if $p$ is prime, then so is $2p + 1$. ...
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2 votes
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Number of "primal" sequences of consecutive numbers of the form $p_1, 2p_2, 3p_3,\dots$ for primes $p_1, p_2,\dots$

I am interested in the number of "primal" sequences of consecutive numbers of the form $p_1, 2 p_2, 3 p_3,\ldots, k p_k$ for primes $p_1, p_2,\ldots, p_k$. For instance, there are $56,157$ ...
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The phenomenon of 'reach' in sieves with regard to prime numbers

In performing the sieve of Eratosthenes, after sieving serially with the first $n$ primes, the smallest composite number remaining is $p_{n+1}^2$. For example, after sieving with $3$, there are no ...
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4 votes
3 answers
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Why is the twin prime conjecture hard?

If $\pi_2(x)$ is the number of twin primes of magnitude less than or equal to $x$. We want to prove that $$\lim_{x\,\to\,\infty}\pi_2(x)=\infty$$ which should be easier than finding and proving an ...
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The distribution of numbers coprime to $p_k\#$ and the implications for the occurrence of twin primes

Consider the interval $[p_k,p_k\#]$. It contains $(\phi(p_k\#))-1=(\prod_{i=1}^k (p_i -1))-1$ numbers relatively prime to $p_k\#$ (NB the $-1$ at the end of each expression is strictly necessary ...
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5 votes
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Goldbach twin and cousin primes

Good day. My question is a counterexample for the following: Is every even number greater than 4 the sum of a number that belongs to set the cousin primes with another number that belongs to the set ...
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Infinite primes history

I am little confused with who was the first to modify Euclid's argument of infinitude of primes from $p_{1}p_{2}...p_{r}+1$ to $p_{1}p_{2}...p_{r}-1$? Some writers say it was E.E. Kummer ,($1878$) (...
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