Yitang Zhang is a largely unknown mathematician who has struggled to find an academic job after he got his PhD, working at a Subway sandwich shop before getting a gig as a lecturer at the University of New Hampshire. He's just had a paper accepted for publication in *Annals of Mathematics*, which appears to make a breakthrough towards proving one of mathematics' oldest, most difficult, and most significant conjectures, concerning "twin" prime numbers. According to the Simons Science News article, Zhang is shy, but is a very good, clear writer and lecturer.

For hundreds of years, mathematicians have speculated that there are infinitely many twin prime pairs. In 1849, French mathematician Alphonse de Polignac extended this conjecture to the idea that there should be infinitely many prime pairs for any possible finite gap, not just 2.

Since that time, the intrinsic appeal of these conjectures has given them the status of a mathematical holy grail, even though they have no known applications. But despite many efforts at proving them, mathematicians weren’t able to rule out the possibility that the gaps between primes grow and grow, eventually exceeding any particular bound.

Now Zhang has broken through this barrier. His paper shows that there is some number N smaller than 70 million such that there are infinitely many pairs of primes that differ by N. No matter how far you go into the deserts of the truly gargantuan prime numbers — no matter how sparse the primes become — you will keep finding prime pairs that differ by less than 70 million.

The result is “astounding,” said Daniel Goldston, a number theorist at San Jose State University. “It’s one of those problems you weren’t sure people would ever be able to solve.”

Unknown Mathematician Proves Elusive Property of Prime Numbers [Erica Klarreich/Wired/Simons Science News]

(*Photo: University of New Hampshire*)

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“It’s one of those problems you weren’t sure people would ever be able to solve.”—–

Or…in my case as a pathetic innumerate ….one of those problems I didn’t even knew to be a problem…and which I couldn’t understand with even the clearest of explanations.

That said, I’m glad its solved. One less problem to worry about.

I think.

Math geek is going to get laid more now.

OK, so that’s at least one number less than 70 million that’ll do that separation trick infinitely often. But …

That’s saying consecutive primes will never, eventually, be separated by more than 70 million. Not quite the same thing.

Which is it? (WIred article no help).

not quite. The second part says you will keep finding pairs separated by less than 70 million, not that you will never find a larger gap.

So 200 million non primes, then a prime, then 60 million non primes, then a prime, then 71 million non primes… condition still satisfied.

As everybody is saying the same thing, I may as well pick on you. SInce you were the first, y’understand. Nothing personal.

I think you’re all getting a tad over-distracted by my use of the phrase

consecutiveprimes and somewhat missing the point that if you can never find a pair of consecutive primes separated by less than 70 million, then you sure as hell ain’t going to find any non-consecutive ones. They would be, perforce, separated by even more. (And I certainly said nothing whatever about never finding a larger gap).I think the issue is whether you read “pairs of primes” as the equivalent of “prime pair” or a not. In Number Theory, a prime pair usually refers to a two consecutive primes. De Polignac’s conjecture is about prime pairs (though it is usually stated in terms of “prime gaps”, which are the differences between consecutive primes).

“That’s saying consecutive primes will never, eventually, be separated by more than 70 million.”

“if you can never find a pair of consecutive primes separated by less than 70 million”

Do you see how you just reversed the condition there? Jeffrey Fisher was responding to your original comment which included that first quote, but in your reply you state the condition as in that second quote.

Let’s consider a few different related statements though just to make sure we’re all clear on what we’re talking about:

A) After a certain point, you can never find primes separated by less than 70 million. (In other words, all pairs of primes > P, consecutive or otherwise, are separated by 70 million or more.)

B) After a certain point, every pair of consecutive primes is separated by less than 70 million.

C) There are infinitely many pairs of primes separated by less than 70 million. (Note that this does not imply that every pair of consecutive primes is less than 70 million apart, just that there’s no limit beyond which there are no more less-than-70-million-apart pairs.)

Note that each of these three statements is making a claim distinct* from the others.

The paper in the article is a proof of C. In your original question, you ask if it’s saying B. Jeffrey Fisher’s response is saying no, it’s C. Then in your response to him you suggest A.

So, to reiterate, no it’s not B, no it’s not A, it’s C.

* For fun, also note that A is incompatible with the other two, that if B is true then C must be true, and that if C is true then B may or may not be true.

Good explanation.

For yet another explanation to Lemoutan: out in the far reaches of prime numbers there might be two consecutive ones that are separated by a billion. There might be billions of subsequent pairs that are separated by billions. But so long as you will always

eventuallyfind another pair separated by the magic N, then the theory is correct.You’re misreading the second statement. It isn’t stating that every pair of consecutive primes is separated by less than 70M. Rather, it’s saying that past any point, there will always be some pair of primes that is within 70M. Other pairs can be further apart.

You can’t get less far apart than consecutive.

The error is in your interpretation using the phrase “consecutive primes”. You can find always find primes separated by less than N.

But not every gap between primes can be less than N. So you might have to jump over a much larger gap between consecutive primes, before you get to another pair separated by less than N.

You may be in error interpreting my interpretation is all I can say. Are you saying the two statements I blockquoted are equivalent?

Actually, what it says is that there is a guarantee that there are infinitely many primes separated by N interval, where N is some number less than 70 million.

It explicitly does not say anything about the number of primes that exist that are separated by intervals larger than N — which would be an interesting area of exploration.

Are there an infinite number of primes separated by N+d, where d is some other interval? Can it be proven there are a finite number of primes separated by N+d? (Highly unlikely).

The real aim of this research is to help move towards a proof of the conjecture that there is an infinite number of twin primes — primes of the form p, p+2 — primes separated by an interval of two.

If a generalisable pattern can be found to how primes occur … well, let’s start by saying it would help us understand physics and the universe and proceed to “all your codes are belong to us”.

The easiest way I’ve seen this written is this:

No matter how far you go on the number line, you will always be able to find two primes (p & q) such that p-q < 70,000,000

Not all primes will satisfy this, but there are an infinite number of prime pairs that do.

What the others replied.

More strongly: there’s an N₀ < 70M so that writing out all the primes in order one sees:

2, 3, 5, 7, 11, prime, …, prime, prime + N₀, prime, …, prime, prime + N₀, prime, … — and that "prime, prime + N₀" pattern never stops appearing.

The reason there's a particular N₀ that works [others might also], is by the pigeon-hole principle: if each separation of 1, 2, 3, 4, 5, …, 70M – 2, 70M – 1 happened only a finite number of times then eventually all primes would be at least 70M apart.

And picking the smallest such N₀, only a finite number of pairs of primes separated by N₀ have a prime in-between, so infinitely often there's a pair separated by exactly N₀ with no other primes in-between those two.

Whilst I fully agree that the

statement says that. (That there’s afirstparticularN (your N0), I don’t agree that the second statement is saying that in another way.The first statement you give is the correct one.

The second is false [and thus not equivalent].

For any gap G, there is a gap in the primes at least that large. E.g. for every g > G, none of the G – 1 consecutive numbers g! + 2, g! + 3, g! + 4, …, g! + G [where g! is 1×2×3×4×⋯(g-1)×g] are prime [the first is larger than 2 but divisible by 2, the second is larger than 3 but divisible by 3, etc].

Oh, I misunderstood your question. My first reply was why your two quoted statements are the same: infinite number of pairs of primes within 70M is equivalent to a particular N₀ < 70M managing it.

But your *summary* of the second statement is incorrect.

It seems apparent that the first statement implies the second, and I think the second statement is only being offered as a consequence of first.

But I think the second statement actually does imply the first. Suppose the second statement were true, but not the first. Then there are infinitely many prime gaps smaller than 70M. But for every particular prime gap G, there are only a finite number of occurrences. (That is, there is no N as in the first statement.) However, since 70M*finite = finite, there would only be a finite number of gaps smaller than 70M, contradicting the second statement.

Yes, and [see my first post] you can go further with a similar argument to show that this occurs for some minimal gap N₀ with consecutive [as in prime and next largest prime] pairs infinitely often.

And while we’re at it, the number of times that gap N₀ occurs with another prime in-between is finite [it contains smaller gaps, and N₀ is the smallest that occurs infinitely often]. So eventually all primes are at least N₀ apart, but ones exactly N₀ apart keep coming up.

I still think the second statement implies what I said about it.

But since I wasn’t as clear as I thought I was, I’ll try putting it another way.

I believe the second statement means that you’ll never have to worry that – eventually – primes will become so sparse that you’ll never again find a pair of primes separated by less than 70 million. In other words that you’ll always be able to find a pair separated by

less than 70 million. Which is nice and comforting in some weird way. N could be all over the shopsome NThe first statement seems – to me – to be about not just any old N but some

N. Nobody claims to know what it is, just that it exists. If it happened to be 2, then Alphonse’s yer uncle.particular“you’ll never again find a pair of primes separated by less than 70 million. In other words that you’ll always be able to find a pair separated by some N less than 70 million”

These two sentences are saying the opposite of each other. Did you mean to write “more” rather than “less” in the first sentence?

In reply to ESRogs, I can only concur that if you omit the “you’ll never have to worry that” part, you will indeed get the contradiction you earnestly seek I suggest you keep it in though. It’s quite important.

No mathematician me (sic – :P) but the writing sure doesn’t seem clear or consistent.

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In 1849, French mathematician Alphonse de Polignac extended this conjecture to the idea that there should be infinitely many prime pairs for any possible finite gap, not just 2.

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… there is some number N smaller than 70 million such that there are infinitely many pairs of primes that differ by N.

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… Zhang is shy, but is a very good, clear writer and lecturer.

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An infinite number of integers (primes) within a finite gap of integers is not possible. The sentence is poorly written and probably not written by Mr. Zhang. The second quote is more understandable and probably more accurate. The “not just 2” has me completely stumped as to its logic. “Pairs” means “2.” ?? Infinite number of gaps of indeterminate size containing two or more prime integers??

Perhaps the first quote should have been written –

‘There is an infinite number of integer sequences (gaps) of some minimum size which each contain at least two prime numbers.’

Is this why Georg Cantor spent so much time in sanatoriums? (sanatoria?)

The gap referred to in the phrase “infinitely many prime pairs for any possible finite gap” is the gap between the primes, not some interval on the number line that all these primes lie between. For example, the gap between 3 and 5 is two, so 3 and 5 are “twin” primes. 5 and 7 are also twin primes, so are 11 and 13. Are there infinitely many pairs of primes that are two apart from each other like that? That’s the open question.

What about other gaps? 7 and 11 are four apart, as are the pairs 13,17 and 19,23, etc. As we go farther along the number line, and look at the gaps between consecutive primes, we start to see some larger and larger gaps, but every now and then we see a small gap as well. So the question is, is there some point at which we never see a small gap anymore?

The paper referred to in the article has shown that as far as you go on the number line, every now and then you will always find a “small” gap between consecutive primes of less than 70 million. Not necessarily that every gap between consecutive primes will be so small, but just that you never get to a point beyond which there will never be gaps that small anymore.

As for the issue of clarity, I’m sure the standard the paper was judged by was how clear it was to fellow mathematicians (note that mathematicians are often highly specialized, so they may take some time understanding each other’s work if they haven’t been studying very similar branches of mathematics). I’m sure the writing assumed knowledge that the layman wouldn’t have, but that fellow mathematicians would. That’s no knock for a technical mathematics paper.

Thanks for the clarification. I think I figured that part out (as I wrote) but that still leaves the “not just 2” puzzle.

That’s beautiful, chains under a certain length linking primes through infinity. Even as a math dolt I can see the elegance.

Something to think about when you’re buying your next sub.

Is there any member of a prime pair that is also a member of another prime pair?

For twin primes there are the pairs 3,5 and 5,7. In fact those are the only ones, because all larger primes are one more or less than a multiple of six, so any twins have to be 6n-1, 6n+1. For more separated pairs there are more options.

I assume* you mean to ask about twin prime pairs. Consider the first few primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

The twin primes within these are:

3 and 5,

5 and 7,

11 and 13, and

17 and 19.So 5 is a member of two twin prime pairs. After that I believe there are no more consecutive pairs of twin primes, because for any set of three integers N, N+2, and N+4, one of them has to be divisible by 3, so they can’t all be prime (except for the case where N=3, which gives the set of 3, 5, 7 mentioned above).* Since any two primes are a “prime pair”, you could just pick any third prime and one of the first two to have a new prime pair and your question would just be asking, “are there three or more primes that exist?”

Guess I left out a constraint, DrDave. Is there another way to pair primes that does not require them to be consecutive? I think then my first question can become a question about tuples.

I think I understand your intent. There is a notion of “prime triplet”, for example, three primes of the form (p, p+2, p+6). For prime triplets, I don’t believe the three have to be consecutive. I’ve seen (5, 13, 17) offered as a triplet of the form (p, p+8, p+12).

Yes, but that adds a constraint that I did not intend. I need to think to a lot more about this. My original question may be trivial.

I think stop commenting here until after I wake up

Question: if it’s not a hype like the “arsenic lifeform!1!!11!” paper – would this help to weaken, or to strengthen public/private key encryption? My understanding is… rudimentary. At best. And maybe even that is hyperbole.

Additional question: Or would the Helfgott paper, cited in the wired piece?

Also, what a name for a mathematician! Yay science! /scnr.

I’ll actually be a little disappointed when the twin prime conjecture is resolved, because since Fermat’s went it’s the only remaining major number-theoretic problem simple enough to explain to non-mathematicians at parties.

Goldbach’s conjecture is easier to explain — every even integer is the sum of two primes.

Goldbach’s is my go-to example, and it’s even easier to explain, but it doesn’t lead the discussion to anything else interesting.

It’s still not resolved. We’re just closer to resolution (if this proof holds up). This paper is a proof that we can never run out of primes that are 70 million apart. It still remains to be proven that we’ll always be able to find primes that are only 1 million apart, or 1 thousand, or 100, or 10, or 2 (this last being the twin prime conjecture).

(Homer Simpson voice) Sounds interesting.

Math is cool and I am glad people are doing it, but I confess to be totally at sea. Good for him, I guess. How about that Eurovision?

Do unknown mathematicians use unknown numbers?

“that there should be infinitely many prime pairs for any possible finite gap, not just 2”

I think you mean any possible *even* finite gap, no?