Yitang Zhang's Bounded Prime Gap Proof

[CraigD]

Recent big news is that an obscure academic, University of NH lecturer (a non-tenure, non-research position) Yitang "Tom" Zhang, has published an important prime number proof, "Bounded gaps between primes" (Annals of Mathematics vol 178). From it’s abstract:

It is proved that

 

[math]lim_{n \to \infty} \mbox{inf}(p_{n+1} - p_n) < 7 \times 10^7[/math]

 

where [imath]p_n[/imath] is the nth prime.

 

For folk like me unfamiliar with it, the [imath]lim \, inf()[/imath] is the lower limit function of a sequence)

 

I can’t find a public copy of the paper, or an affordable way to access it at the Annals of Math website. But looking only at its abstract, have a very different reaction than the one expressed in most news articles, such as this one, that this is most importantly “a step towards solving the twin primes conjecture”, and unsurprising. My reaction is of shock, because at a glance, the idea that (to put Zhang’s paper’s abstract in words) the prime gap [imath]g_n = p_{n+1} - p_n[/imath] approaches a finite number (less than 70,000,000) as [imath]n[/imath] approaches infinity contradicts my intuitive understanding of prime number theorem, and directly, Rankin’s 1997 proof that

 

[math]g_n > \frac{c\log n\log\log n\log\log\log\log n}{(\log\log\log n)^2}[/math]

 

Am I the only one feeling this sense of shock at Zhang’s proof’s conclusion?

[phillip1882]

i'm pretty sure the proof is false.

all analysis of the distribution of primes point to the prime gap increasing indefinitely.

 

there should be a prime gap of 70,000,000 somewhere between e^70000000 and e^70000001.

[phillip1882]

we can use the Chinese remainder theorem to prove that there is no minimum gap between primes.

consider an arbitrary list of 70,000,000 primes.

i prove that you can contruct a set of composite numbers such that n, n+1, n+2... n+70,000,000 is all composite.

let's do so with a set of 7 primes.

x = 0 mod 2

x+1 = 0 mod 3

x+2 = 0 mod 5

x+3 = 0 mod 7

x+4 = 0 mod 11

x+5 = 0 mod 13

x+6 = 0 mod 17

which is equavalent to...

x = 0 mod 2

x = 2 mod 3

x = 3 mod 5

x = 4 mod 7

x = 7 mod 11

x = 8 mod 13

x = 11 mod 17

using the Chinese remainder theorem; we have...

0*255255*1 +2*170170*1 +3*102102*3 +4*72930*2 +7*46410*1 +8*39270*4 +11*30030*15 = 8,379,158.

let's test.

8379158/2 = 4189579

8379159/3 = 2793053

8379160/5 = 1675832

8379161/7 = 1197023

8379162/11 = 761742

8379163/13 = 644551

8379164/17 = 491892

so. you can use this for any number of primes.

[CraigD]

Cool proof :thumbs_up Though I’m an easy audience – I think that all elementary proofs are cool :)

 

all analysis of the distribution of primes point to the prime gap increasing indefinitely.

I agree.

 

i'm pretty sure the proof is false.

I’ve a smoldering suspicious Zhang’s proof is wrong, but it’s not credible that the peer reviewers at Princeton U’s Annals of Math, or any credible jounal, would miss something as obvious as your criticism, Phillip.

 

I think, like me, you don’t have an immediate familiarity with the [imath]\mbox{lim}_{n \to \infty} \mbox{inf}(x_n)[/imath] limit inferior sequence function in term of which Zhang’s proposition is stated. The function returns the least value of the sequence for all terms greater than the n-th, where [imath]n \ge m[/imath] where [imath]m \to \infty[/imath], so in natural language, the proposition is stating that, no matter how great the primes, there will always be a prime gap less than 70,000,000. As you’ve clearly shown, there will be prime gap greater than that – that’s not the question implicit in Zhang’s work. It’s not about the average density of primes, but about the “lumpiness” of their distribution. If we were talking not about integers on the number line, but raisins in expanding bread dough (an analogy usual in cosmology, not math), the question would be, do we ever get raisins stuck close together, no matter how much the dough expands?

[phillip1882]

so in natural language, the proposition is stating that, no matter how great the primes, there will always be a prime gap less than 70,000,000

okay that makes more sense. that's a hypothesis i can accept, and agree to.

the prime gap g_n = p_{n+1} - p_n approaches a finite number (less than 70,000,000) as n approaches infinity.

this is what i objected to. I'm pretty sure the prime gap does not approach a finite number. there will be infinitely many prime gaps of any size. the larger you go, the "fewer" they will be, but you'll always have infinitely many of them.

[C1ay]

I'm shocked as well. My intuitive feeling is that the gap would approach infinity as n approaches infinity, i.e. that the string of composites between primes would approach infinity as n approaches infinity.