Is the number of twin primes really infinite?

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

A very timely post indeed.

This paper: http://arxiv.org/abs/math.NT/0405509
seems to prove that it is infinite. (very new paper)

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

It's been proven wrong :D

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 !

How horrible must that feel, to have to withdraw such a proof =[

What was the error in the proof? They took the paper down...