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Progress on the Twin Prime Conjecture

The Chinese mathematician Yitang Zhang made great strides in one of the big problems in number theory called the Twin Prime Conjecture. He did not solve it but proved a related result that is considered quite remarkable — and he got his paper accepted for publication in the very prestigious math journal Annals of Mathematics.

If you wish you can read the sources below regarding the news on this, but I’d like to say things my way.

First, let’s explain some things. The ‘numbers‘ we’re talking about here are mainly positive whole numbers 1, 2, 3, 4, …, (which we call positive integers, or natural numbers). (So we’re not talking fractions here.)

A number is prime when it is not divisible by any other number beside 1 and itself. So for example, 5 is a prime, while 6 is not prime (we call it composite, since it is divisible by 2 and 3, beside 1 and 6).

So the sequence of prime numbers starts off and proceeds like this:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, …

and they go on and on forever. We know that there are infinitely many prime numbers (and the proof is not hard).

Now if you look closely at this list of primes, you notice something interesting. That there are ‘consecutive’ pairs of primes. Ones whose difference is 2.  Like the pair 5, 7, and the pair 11, 13, and the pair 29, 31, etc etc. These are called twin primes.

The Twin Prime Conjecture says that there are infinitely many of these twin pairs of primes.

But unlike Euclid’s proof that there are infinitely many primes (which is easy), the proof of the Twin Prime conjecture is extremely hard or just unknown. (Or maybe it’s not true!) No one has been able (so far) to settle the question or find a proof of it.

Since that is too hard, mathematicians try to ask and maybe settle simpler, but related, questions that they may have a chance at answering.

For example, instead of looking at pairs of primes whose difference is 2, why not look at ones whose difference is some number L?

As a related conjecture, one can ask:

Q: Is there a number L such that there are infinitely many pairs of primes whose difference is no more than L?

When L = 2, this question reduces to the Twin Prime Conjecture. For L = 10, let’s say, we can have pairs like 7, 11, whose difference is 4 which is no more than 10. Another pair would be 17, 23, whose difference is 6 (again no more than 10). Etc.

Now what Professor Yitang Zhang proved is that the answer to question Q is YES! There is such a fixed difference L. He proved that if you take L to be 70,000,000 (70 million) then there are infinitely many pairs of primes whose difference is no more than 70 million.

(So in particular we don’t know if there infinitely many pairs of primes whose difference is not more than 100 (or 1000, or even a million). These would still be interesting but apparently quite hard to answer.)

Once again, we have very simple sounding questions (a babe can ask them!) that are very hard to prove, and indeed we do not even have proofs for them.

REFERENCES:

These two reference are short reports on the result.

Nature MagazineNew Scientist

The following reference is slightly longer and more detailed:

Simon’s Foundation

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  1. June 15, 2013 at 2:12 am

    Hello,
    I have to submit a new theory on the distribution of prime numbers
    excuse me, because, I don’t speak anglish correct
    solution distribution prime numbers here :

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