# 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 ...
1 vote
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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 ...
1 vote
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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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### $H(n)=\lfloor\dfrac{b}{n}\rfloor- \lfloor \dfrac{a}{n} \rfloor=$ (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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### 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 ...
143 views

### 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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### 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 ...
1 vote
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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 ...