Questions tagged [twin-primes]
For questions on prime twins.
314 questions
1
vote
1
answer
84
views
An equivalent formulation for the twin prime conjecture
Consider Oeis $A065387$: $a(n)=\sigma(n)+\phi(n)$, where $\sigma(n)$ represents the sum of all divisors of $n$ and $\phi(n)$ is Euler's totient function.
I ask to prove that the assertion
the ...
- 3,064
4
votes
1
answer
180
views
Are the sums of every prime-pair equal to the sum of two previous prime-pairs?
If you sum any two twin-primes, starting with $(17,19)$ as the smallest valid pair, their sum will always be equal to the sum of two previous twin-prime sums, not necessarily consecutively though.
I'm ...
- 51
3
votes
0
answers
106
views
Extending the twin‑prime polynomial Diophantine pair $(n^2+p,n^2+p+2)$ to a triple
While playing with the identity
$$
p(p+2)+1=(p+1)^2,
$$
I noticed that for any integer $n$ and any integer $p$ (twin prime not actually needed here),
$$
(n^2+p)(n^2+p+2)+1=(n^2+p+1)^2.
$$
So the pair
$...
- 31
10
votes
1
answer
841
views
Prove that a bivariate quartic polynomial can never be a perfect square
This question arose when trying to answer a broader question posted on SE:
$n+1$ and $n \phi(n)+1$ are both perfect squares if and only if $n$ is a product of twin primes?
The question below is a ...
- 581
1
vote
0
answers
104
views
Something like primality test by checking if the expression is a square only [duplicate]
Let
$a(n)$ be A002145, i.e., an integer sequence known as a sequence of primes of the form $4k+3$.
$b(n)$ be A037074, i.e., an integer sequence known as a sequence of numbers that are the product of ...
user929945
0
votes
1
answer
91
views
Statistical patterns in primes generated from twin prime offsets
I analyzed primes generated by applying four linear offsets to the first element of twin prime pairs ( 𝑝 , 𝑝 + 2 ), for 𝑝 ≤ 1,000,000:
𝑞1 = 2𝑝 + 1, 𝑞2 = 2𝑝 + 7, 𝑞3 = 2𝑝 − 3, 𝑞4 = 2𝑝 + 3.
...
2
votes
0
answers
86
views
On zeroes of a prime twin function
Let
$$ TH(x) = \sin^2(\pi x) + \sin^2 \left( \frac{\pi \cdot (\Gamma_h(x)+1)}{ x}\right) + \sin^2 \left( \frac{\pi \cdot (\Gamma_h(2+x)+1)}{ 2 +x}\right).$$
Where $\Gamma_h$ is Hadamard’s Gamma ...
- 18.4k
2
votes
1
answer
169
views
Twin prime from Cramér's model
The strong form of the twin prime conjecture states
$$\#\left\{p\le x : p,\, p+2k \text{ are both prime}\right\}= C_{2k}\operatorname{Li}_2(x) + \mathcal O_{k,\varepsilon} \left(x^{\frac 12 +\...
- 11.1k
1
vote
1
answer
311
views
Prime triplets table
Does anyone know where I can find a table showing the number of prime triplets less than or equal to $x$, indexed by powers of $10$, up to $10^{18}$?
I need it to see how well an estimator ...
- 291
1
vote
1
answer
114
views
Generalization of the fact that all sufficiently large twin-prime (averages) are of the form $6x \pm 1$, a conjecture.
Put $0 \in \Bbb{N}$. I came up with this conjecture using SymPy and a simple program that exhaustively checks. But the statement requires formal proof.
Conjecture. For each $n \in \Bbb{N}$, the ...
- 22.7k
3
votes
0
answers
75
views
Twin-prime "circular locks" modulo $p_n\#, n\geq 1$ have symmetric colorings even for any other unrelated prime $q \nmid p_n\#$! (Picture inside)
All twin-primologists know that a number $x \in [p_n+2, p_{n+1}^2 - 1)$ (sieve of Eratosthenes bounds) is a twin prime average (the number between the two primes in a twin prime pair) precisely when $...
- 22.7k
0
votes
1
answer
39
views
Additive $(+1, +1, \dots)$ paths in divisibility by prime numbers $p_1, \dots, p_n$ for fixed $n \geq 1$ have associated symmetries.
Consider the divisors of $p_n\# = 30$ for $n = 3$. You can either write them as vectors $(1,0,1)$ meaning $2 \mid x$ and $5 \mid x$ but $3\nmid x$ or simply as a number ($10 = 2\cdot 5$ in this case) ...
- 22.7k
0
votes
1
answer
96
views
Relating to $6xy \pm x + \pm y$ formulation of the twin prime conjecture and the apparent monoids formed in $\Bbb{Z}$.
Lemma. $*: \Bbb{Z}^2 \to \Bbb{Z}$ given by $x*y :=6xy + x+y$ and similarly $x\star y := -6xy + x + y$ define isomorphic monoid structures. The isomorphism is the $\eta(z) =-z$ involution.
Definition....
- 22.7k
2
votes
1
answer
107
views
A Conjectured Formula for the Average Gap Between Twin Primes Below $10^{N}$
I noticed a pattern in the average gap between consecutive twin primes less than $10^N$. Here's the formula.
Average gap between consecutive twin primes =
$\begin{cases}
(2N - 1)^2 + 1, & \text{...
3
votes
0
answers
2k
views
Proving primes and twins using CRT [closed]
I'm not experienced with writing papers or proofs, so I hope you will me correct my work, or point out where I might be going wrong. At any rate, I'm happy to have this out of my head. Thank you for ...
- 31
14
votes
1
answer
312
views
Is it possible to map infinitely many primes to infinitely many other primes with a linear function?
I was wondering if this is even posible because I can not see an answer...
Given a subset $S\subset P$ from the set of prime numbers $P$. Let $S$ have an infinite number of elements. Is there any ...
- 141
2
votes
1
answer
124
views
Brun's sieve: There are infinitely many primes $p$ such that $p+2$ has at most $20$ prime factors
This is exercise 2 (p. 120) of chapter 6 in Iwaniec-Kowalski's book "Analytic number theory". More specifically, let $\pi_2(x, z)$ denote the number of primes $p\leq x$ for which $p+2$ has ...
- 588
1
vote
2
answers
124
views
Can Dickson's conjecture with $b_i=1$ be proven for one $n,$ given that there are no obvious divisibility restrictions preventing this from happening?
I was reading this answer to it's question, and came across Dickson's Conjecture, because I was independently investigating the case where $b_i=1$ for $i\in\{1,2,\ldots,k\}.$
Dickson's conjecture says ...
- 25.2k
1
vote
0
answers
67
views
Limit kernel maybe interesting supposing opposite of twin prime conjecture (By way of contradiction).
Define. $G_n =\{ x \in \Bbb{Z} : x^2 = 1 \mod n\}$ for any $n \geq 1$. Clearly, moduloing everything by $n$, it is a subgroup of $\Bbb{Z}/n^{\times}$ and has size $2^{\omega(d) - [2\mid n]}$, where ...
- 22.7k
0
votes
1
answer
69
views
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 ...
- 45
1
vote
1
answer
165
views
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 ...
- 71
0
votes
1
answer
77
views
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} \...
- 22.7k
6
votes
1
answer
518
views
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 ...
- 3,673
0
votes
1
answer
175
views
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 ...
- 4,777
2
votes
0
answers
175
views
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 ...
- 4,777
2
votes
1
answer
206
views
$f(x) = \left\lfloor\frac{x- a}{b} \right\rfloor$ 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 ...
- 22.7k
1
vote
0
answers
67
views
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. $...
- 7,731
2
votes
2
answers
229
views
Distribution of primes in primitive Pythagorean triples
My Observation:
I've observed a pattern where for every pair of twin primes ($p$, $p+2$), there appears to be at least one primitive Pythagorean triple ($a$, $b$, $c$) such that one of the twin primes ...
0
votes
0
answers
114
views
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 ...
- 55
1
vote
0
answers
147
views
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 ...
- 2,852
3
votes
1
answer
206
views
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 $n$ ...
- 2,852
1
vote
4
answers
176
views
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 ...
- 13
3
votes
0
answers
141
views
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 ...
- 18.4k
2
votes
2
answers
158
views
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 $1$ ...
- 18.4k
4
votes
0
answers
254
views
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 $A$ ...
1
vote
1
answer
91
views
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 ...
- 18.4k
11
votes
1
answer
686
views
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:
$$ \lim\limits_{n\to \infty} \frac{\pi^2(n)}{n\pi_2(n)} = 0.7550363087870907 \cdots\cdots (1) $$...
- 2,852
1
vote
1
answer
82
views
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 ...
- 22.7k
22
votes
2
answers
710
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 $n \phi (n) + 1$ ...
- 581
0
votes
1
answer
82
views
$A(x) = \prod_{p,q \in \Bbb{P}} (1 - \frac{x^2 - 1}{pq}) = 0$ 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 ...
- 22.7k
1
vote
1
answer
338
views
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} \...
- 2,575
0
votes
0
answers
252
views
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 ...
- 7,731
1
vote
1
answer
95
views
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 ...
- 45
2
votes
2
answers
278
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,\...
- 7,731
1
vote
0
answers
102
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 ...
- 11
0
votes
1
answer
115
views
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 ...
- 9,768
0
votes
1
answer
94
views
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) ...
- 65
0
votes
1
answer
82
views
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 $$f(x) = \sum_{d \mid p_{\...
- 22.7k
0
votes
0
answers
79
views
Number of divisors of composite number between and adjacent to twin primes
I am investigating properties of the number of divisors of composites between and adjacent to twin primes. When running some numeric calculations in Python (which are hopefully correct) I get the ...
- 559
0
votes
0
answers
84
views
The series $\sum \frac{1}{p_iq_i}$ where $p_i,q_i$ are twin primes
I am interested in the series comprising the inverses of the products of twin prime pairs: $$\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 ...
- 7,731