# First Proof That Infinitely Many Prime Numbers Come in Pairs

A U.S. mathematician claims a breakthrough toward solving a centuries-old problem

Mathematician Yitang Zhang has outlined a proof of a "weak" version of the twin prime conjecture. Image: Maggie McKee

From Nature magazine

Cambridge, Massachusetts

It’s a result only a mathematician could love. Researchers hoping to get ‘2’ as the answer for a long-sought proof involving pairs of prime numbers are celebrating the fact that a mathematician has wrestled the value down from infinity to 70 million.

“That’s only [a factor of] 35 million away” from the target, quips Dan Goldston, an analytic number theorist at San Jose State University in California who was not involved in the work. “Every step down is a step towards the ultimate answer.”

That goal is the proof to a conjecture concerning prime numbers. Those are the whole numbers that are divisible only by one and themselves. Primes abound among smaller numbers, but they become less and less frequent as one goes towards larger numbers. In fact, the gap between each prime and the next becomes larger and larger — on average. But exceptions exist: the ‘twin primes’, which are pairs of prime numbers that differ in value by 2. Examples of known twin primes are 3 and 5, or 17 and 19, or 2,003,663,613 × 2195,000 − 1 and 2,003,663,613 × 2195,000 + 1.

The twin prime conjecture says that there is an infinite number of such twin pairs. Some attribute the conjecture to the Greek mathematician Euclid of Alexandria, which would make it one of the oldest open problems in mathematics.

The problem has eluded all attempts to find a solution so far. A major milestone was reached in 2005 when Goldston and two colleagues showed that there is an infinite number of prime pairs that differ by no more than 16. But there was a catch. “They were assuming a conjecture that no one knows how to prove,” says Dorian Goldfeld, a number theorist at Columbia University in New York.

The new result, from Yitang Zhang of the University of New Hampshire in Durham, finds that there are infinitely many pairs of primes that are less than 70 million units apart without relying on unproven conjectures. Although 70 million seems like a very large number, the existence of any finite bound, no matter how large, means that that the gaps between consecutive numbers don’t keep growing forever. The jump from 2 to 70 million is nothing compared with the jump from 70 million to infinity. “If this is right, I’m absolutely astounded,” says Goldfeld.

Zhang presented his research on 13 May to an audience of a few dozen at Harvard University in Cambridge, Massachusetts, and the fact that the work seems to use standard mathematical techniques led some to question whether Zhang could really have succeeded where others failed.

But a referee report from the Annals of Mathematics, to which Zhang submitted his paper, suggests he has. “The main results are of the first rank,” states the report, a copy of which Zhang provided to Nature. “The author has succeeded to prove a landmark theorem in the distribution of prime numbers. … We are very happy to strongly recommend acceptance of the paper for publication in the Annals.”

Goldston, who was sent a copy of the paper, says that he and the other researchers who have seen it “are feeling pretty good” about it. “Nothing is obviously wrong,” he says.

For his part, Zhang, who has been working on the paper since a key insight came to him during a visit to a friend’s house last July, says he expects that the paper’s mathematical machinery will allow for the value of 70 million to be pushed downwards. “We may reduce it,” he says.

Goldston does not think the value can be reduced all the way to 2 to prove the twin prime conjecture. But he says the very fact that there is a number at all is a huge breakthrough. “I was doubtful I would ever live to see this result,” he says.

Zhang will resubmit the paper, with a few minor tweaks, this week.

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1. 1. klarg 01:30 AM 5/15/13

Give that man a platinum pocket protector!

2. 2. Dr. Science 12:38 PM 5/15/13

I don't see what the big deal is here. I've already come up with a truly elegant generalization of this proof that extends the twin concept to the set of all integers, not just the primes. But I don't have room to write down the proof in the margin of this comment.

3. 3. Proventus 05:26 PM 5/15/13

At some point, wouldn't the separation restart?

4. 4. remple 05:47 PM 5/17/13

I love Dr. Science's note, and anxiously await formal publication of his or her astounding result.

5. 5. joelwmson 06:39 PM 5/18/13

On the other hand it is easy to prove that there are an infinite number of prime pairs separated by more than 70 million: Let p(i) be the ith prime; PN the product of the first N primes:

PN = p(1)*p(2)*...p(N)

Then PN+1 is either prime or divisible by some p(M), M>N

Whole numbers from PN+2 to PN+p(N) are all non-prime, so:

For all p(N) > 70 million there are prime pairs separated by more than 70 million non-primes.

This contradicts the statement in the article: "Although 70 million seems like a very large number, the existence of any finite bound, no matter how large, means that that the gaps between consecutive numbers don’t keep growing forever."

6. 6. Stapleton 04:29 PM 5/23/13

The sentence beginning " Whole numbers"is not correct.

7. 7. JeremyHorne 06:52 PM 5/23/13

I find it fascinating that people use the incorrect expression "infinite number", when the proper verbiage is simply "infinite". I know that most people use this phrase, but I see issues. There surely are infinite numberS, but, in this dimension at least, I seriously doubt if there is an infinite number. "Number" refers to a specific quantity, but how could there be an infinite set quantity, for such would be unknowable. Otherwise put, "number" is a discrete pairing with a quantity, whereas pairing a discrete quantity (number) to "infinity" is impossible. "infinity" is a range of values extending endlessly. A search under "correct to say infinite number?" will amplify the discussion. Shortly put in all this, infinity is not a number. I think the posting by Mark C. Chu-Carroll on 13 October 13 2008 at http://scienceblogs.com/goodmath/2008/10/13/infinity-is-not-a-number/ pretty much says it all.

8. 8. Willington 08:18 PM 5/23/13

I am not happy with the stated definition of a prime number. The preferable definition is an integer with two and only two distinct factors. The definition given in the article allows for 1 to be a prime number and this is not true.

9. 9. europamoon100 09:06 PM 5/23/13

@Willington: You are just being difficult for no good reason. The ususal definition of a prime number is "a positive whole number that is divisible only by 1 and by itself. However, the number 1 is a special case, and by definition, it is not a prime number."

The primary reason for definining 1 as being a nonprime is to make the unique-factorization theorem of the positive integers work out. For example 60 = (2)(2)(3)(5), uniquely. If 1 were allowed to be a prime number, then another way of factoring 60 would be 60 = (1)(1)(2)(2)(3)(5), and there would be many more.
The unique factorization of the polynomials, which was first proven by Gauss, would also have problems if 1 were allowed to be a prime factor.

Note that when Gauss proved this theorem, he made some assumptions that he should not have, most notably that all polynomials of an odd degree have at least one real root. YES, this is obvious, but Gauss whould have proven it. Other mathematicians did later on.
D.A.W.

10. 10. europamoon100 09:08 PM 5/23/13

Everyone needs to realize that "Dr. Science" was only joking with his comment.

11. 11. europamoon100 09:20 PM 5/23/13

In the article, there is a statement that is obviously incorrect - and that came from sheer carelessness.
The statement says "...the existence of any finite bound, no matter how large, means that that the gaps between consecutive numbers don't keep growing forever."

The gaps between any two CONCEIVABLE consecutive numbers is 1, in other words, unity. So, what's the big deal.

If the writer, Maggie McKee, had meant "the gaps between consecutive prime numbers", then she should have written so. Another way to put it would be "the gaps between consecutive primes", which would be shorter than the original version, but more meaningful.

12. 12. europamoon100 09:37 PM 5/23/13

@joelwmson: Joel has gotten his inequalities backwards. He stated "...it is easy to prove that there are an infinite number of prime pairs separated by more than 70 million..."
The whole goal of the theorem was to prove that it was true for pairs of primes separated by LESS THAN 70 million."
As you mentioned, to prove the theorem for MORE THAN 70 million is simple and obvious.

Also, the next goal after this theorem passes close scrutiny (if it does), will be to cut that number 70 million (an upper bound) down to smaller and smaller numbers. To cut it down to a number that you could count on your fingers and toes seems to be VERY dubious now.

Also, in a minor point in English grammar, "that there are an infinite number of prime pairs", should be replaced by "that there is an infinite number of prime pairs".
Believe it or not, the concept of "infinity" or "an infinite number" is grammatically singular.

So many people do not realize that all of these are gramatically singular: {couple, duo, trio, quartet, quintet, a dozen, a score, a hundred, a thousand, a million}.
"The Dynamic Duo is riding the Batmobile into Gotham City," and NOT "the Dynamic Duo are riding the Batmobile into Gotham City."
"A score of sinister horsemen is raiding the Batcave," and not "are raiding the Batcave."

13. 13. sflandherr 12:26 AM 5/24/13

@Stapleton
The sentence beginning " Whole numbers"is indeed correct.
Consider the divisibility of (k-PN) where k is an integer in the range PN+2 to PN+p(N) inclusive.

14. 14. Christofurio 04:06 PM 5/25/13

Yes, Europa Moon, I think we got the joke.

- Andrew Wiles.

15. 15. Cauchy in reply to joelwmson 12:25 PM 5/26/13

An upperbound is very valuable, a lower bound is useless. If I ask you to prove for example that an integral like Int(f(x))[0,infinity] exist, good luck doing it with a lower bound.

16. 16. slowhands 07:27 PM 5/26/13

That is so humorous I was almost tempted to smile.

17. 17. joelwmson 10:26 PM 5/26/13

Stapleton: Not much of a comment w/o justification.

Given K: 2(=p(1)) <= K <= p(N), then

K is either a prime number, or, K is divisible by two or more of the prime numbers in the set (p(1), p(2), ..., p(N))

PN is divisible by all of the primes in that set, so,

PN + K is non-prime for each value of K.

So for all p(N) > than 70 million there are infinite prime pairs separated by more than 70 million non-primes, since there are infinite primes > 70 million. (There JeremyHorne, does that roll off the tongue better? :-) )

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