Is the number of twin primes really infinite?

In summary, the conversation discusses the conjecture of infinite twin primes and the methods used to prove it. The question of why the proof has not been found yet is also raised. Mention is made of Brun's Theorem and other related conjectures. Finally, a link to a new paper is shared, but it is later revealed to be incorrect.
  • #1
maverick280857
1,789
5
Hi

I've been wondering...the conjecture which states that the number of twin primes is infinite has neither been proved nor disproved so far. We know that the number of primes is infinite and I have come across two methods of proving this.

My question is: why can't we actually prove that the number of twin primes, i.e. the number of distinct pairs of the form

(p, p+2)
or
(p-2, p)

where both members of the ordered pair are prime, is infinite? If we assume that the number is finite, would we reach an absurdity? If yes, then reductio-ad-absurdum should be the method of proof. Why then is it that no convincing methods have been proposed to prove this conjecture (or disprove it) for so many years?

Brun's Theorem (http://mathworld.wolfram.com/BrunsConstant.html) describes (perhaps not as rigorously as we would like) the scarcity of twin primes. There are conjectures of all kinds related to twin primes and they are indeed, quite interesting...

Cheers
Vivek
 
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  • #3
Hi

Thanks so much for this link...its very interesting...and captivating (just like math and science are in general). I do not know enough number theory yet to understand some techniques in this paper but I am learning and so hope to read this in depth sometime soon.

Cheers
Vivek
 
  • #4
It's been proven wrong :D
 
  • #5
AmirSafavi said:
It's been proven wrong :D

I hope you are referring to the proof in your previous post, and not the twin-prime conjecture itself !
 
  • #6
How horrible must that feel, to have to withdraw such a proof =[
 
  • #7
What was the error in the proof? They took the paper down...
 

FAQ: Is the number of twin primes really infinite?

What are twin primes?

Twin primes are pairs of prime numbers that differ by 2. Examples include (3, 5), (11, 13), and (41, 43).

How do we know that there are infinitely many twin primes?

This is based on a conjecture called the Twin Prime Conjecture, which states that there are an infinite number of twin primes. It has not been proven yet, but there is evidence to suggest that it is true.

How can we find twin primes?

There is no known formula for finding twin primes, so the only way to find them is by checking each pair of consecutive odd numbers to see if they are both prime. This process becomes increasingly difficult as the numbers get larger.

Are there any patterns in the distribution of twin primes?

There have been attempts to find patterns in the distribution of twin primes, but none have been proven. Some mathematicians believe that there is no discernible pattern, while others continue to search for one.

Why is the infinite nature of twin primes important?

The infinite nature of twin primes is important because it helps us understand the nature of prime numbers and their distribution. It also has applications in cryptography and number theory. Additionally, the search for twin primes has led to the discovery of new mathematical techniques and algorithms.

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