Does this imply infinite twins?

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In summary, the twin prime counting function has this form: $\pi_2(n)=f(n)+\pi(n)+\pi(n+2)-n-1,$ where $f(n)$ is the number of twin composites less than or equal to n.
  • #1
e2theipi2026
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I can prove the twin prime counting function has this form:

[tex]\pi_2(n)=f(n)+\pi(n)+\pi(n+2)-n-1,[/tex]

where [tex]\pi_2(n)[/tex] is the twin prime counting function, [tex]f(n)[/tex] is the number of twin composites less than or equal to [tex]n[/tex] and [tex]\pi(n)[/tex] is the prime counting function.

At [tex]n=p_n,[/tex] this becomes

[tex]\pi_2(p_n) = f(p_n) + \pi(p_n) + \pi(p_n + 2) - p_n - 1.[/tex]

With this form, can I make the following argument?: Assume the twin prime counting function becomes a constant [tex]c[/tex], then I can change the twin prime counting function to [tex]c[/tex] in the equation. The prime counting function [tex]\pi(n)[/tex] at the prime sequence [tex]p_n[/tex] is just [tex]n[/tex], so I can change that to [tex]n[/tex]. Because I'm assuming no more twin primes, [tex]p_n+2[/tex] is not a prime so [tex]\pi(p_n+2)[/tex] will also become [tex]n[/tex], the equation directly above this paragraph can therefore be simplified to:

[tex]c = f(p_n) + 2n - p_n - 1.[/tex]

Adding [tex]1[/tex] to both sides of this and rearranging it gives,

[tex]p_n - f(p_n) = 2n - b[/tex], where [tex]b=c+1.[/tex]

The right side of [tex]p_n - f(p_n) = 2n - b[/tex]

has only one possible parity, either odd or even because it is an even number [tex]2n[/tex] minus a constant [tex]b.[/tex]

But, the left side can be both odd and even many times over because [tex]f(p_n)[/tex] can be odd or even and is subtracted from [tex]p_n[/tex] which is odd for [tex]p>2.[/tex]

So, the left side will change parity for different values of [tex]n,[/tex] while the right side of the equation will remain one parity. Therefore, the two sides cannot be equal for all [tex]n.[/tex]

This seems to show the twin prime counting function cannot become constant and therefore, there are infinite twin primes. Now assuming I can prove the form of the twin prime counting function given at the beginning of this question, does that argument hold water?
 
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  • #2
What is the function $f(n)$?
 
  • #3
Amer said:
What is the function $f(n)$?

[tex]f(n)[/tex] was defined at the beginning of the post. If you mean more detail, it is counting the number of "smaller" twin composites [tex]\le n.[/tex] So, it is counting all composite [tex]k\le n[/tex] such that [tex]k+2[/tex] is also composite.
It would seem what I have shown is that the Twin Prime Conjecture is equivalent to proving that [tex]f(p_n)[/tex] changes parity an infinite number of times. Any ideas? :)
 
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FAQ: Does this imply infinite twins?

What is the concept of "infinite twins" in science?

"Infinite twins" is a theoretical concept in science that suggests the possibility of multiple versions of ourselves existing in parallel universes or alternate timelines. These versions, known as "twins," could potentially be identical or slightly different from ourselves, depending on the theory.

How does the idea of infinite twins relate to the theory of parallel universes?

The idea of infinite twins is closely related to the theory of parallel universes, which proposes the existence of multiple universes that exist alongside our own. In this theory, each universe would have its own set of physical laws and could potentially contain versions of ourselves, including infinite twins.

Is there any scientific evidence to support the existence of infinite twins?

Currently, there is no scientific evidence to support the existence of infinite twins or parallel universes. These concepts are purely theoretical and are based on mathematical equations and theories, such as quantum mechanics and string theory.

Can the concept of infinite twins be tested or proven?

As of now, there is no way to test or prove the existence of infinite twins or parallel universes. These concepts are still being explored and debated in the scientific community, and there is much more research and experimentation needed before any definitive conclusions can be made.

What are the implications of infinite twins if they do exist?

If infinite twins were to exist, it would have significant implications for our understanding of the universe and our place within it. It could potentially challenge our perception of reality and the concept of free will, as well as open up new avenues for scientific exploration and discovery.

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