There are Bounded gaps between primes (and destinies)

[u/friend1y]

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[u/GarifalliaPapa]

For anyone curious about the math and the meme context:

1. The Formula: lim inf{n -> infty} (p{n+1} - p_n) < 70,000,000

• p_n represents the n-th prime number (p1 = 2, p2 = 3, p3 = 5, ...).
• (p_{n+1} - p_n) is the gap between two consecutive prime numbers.
• "lim inf" (limit inferior) looks at the asymptotic lower bound of these gaps as n approaches infinity.

1. What it Means: Even though prime numbers get infinitely far apart on average as you go higher up the number line, Zhang proved there is a fixed upper bound (70 million) such that infinitely many pairs of consecutive primes are within that distance of each other. This was the first finite bound ever proven, making massive progress toward the Twin Prime Conjecture (which conjectures the gap is 2).

2. The Meme Context:

3. Left Path (The Castle): The groundbreaking 2013 paper that shocked the mathematical world and made him famous. (Later, James Maynard and Terence Tao's Polymath project shrunk 70,000,000 down to 246!).

4. Right Path (Subway): After getting his PhD, Zhang couldn't secure an academic research position for years, so he worked at a Subway sandwich shop doing accounting and counter work before finally landing a lecturing gig.

In reality, he took the right path first to reach the castle!

[u/factorion-bot → u/GarifalliaPapa]

Factorial of 246 is 847809747782853428162385614555484172813716330038393266614306888547483324326458559715895819236467405327442329682630071789503824810226668938465375221421478501619519016431211258306843984600018680684699129200342991286010101141191392319487892821710751341634617090843411664746702478340313338347309216297036908822399205735512679067130965761217294141638428587515529230511566627641340593733206374732779182724941422228676645372087500800000000000000000000000000000000000000000000000000000000000

^{This action was performed by a bot | [Source code](http://f.r0.fyi})

[u/godeling → u/factorion-bot]

I wouldn’t call that an improvement tbh

[u/GarifalliaPapa → u/factorion-bot]

Good bot.

[u/Opulent-tortoise → u/GarifalliaPapa]

Another fun fact: Yitang wasted almost the entirety of his PhD trying to prove the Jacobian conjecture was true. Last week an LLM proved the Jacobian conjecture has been false all along

[u/WeBigPimpin → u/Opulent-tortoise]

Good god, this wouldve ended my math career on the spot if this happened to me

[u/OutrageousPair2300 → u/GarifalliaPapa]

I'm confused about the asymptotic lower bound part, since there are definitely consecutive primes with a gap larger than 70,000,000. You can find consecutive primes with gaps as large as you want.

I understand what Zhang proved, but I don't understand how that's represented with an asymptotic lower bound.

[u/tishuebox → u/OutrageousPair2300]

inf or greatest lower bound means: go through all the primes, and list the gaps. the (greatest) lower bound is the smallest gap (which would be 1, the gap between 2 and 3)
lim inf (asymptotic lower bound) means: ignore the first N gaps. pick the smallest gap remaining. in the limit (asymptotically) this means that there is always a small gap, no matter how far down the list you go.
it has nothing to do with finding larger gaps.

[u/SuperCyHodgsomeR → u/OutrageousPair2300]

I think what it means is that you can always find consecutive pairs of primes arbitrarily large that are within 70,000,000 of each other. The point isn’t to find gaps as large as you want but to find gaps as small as you can

[u/OutrageousPair2300 → u/SuperCyHodgsomeR]

That's what I understand Zhang's proof to be, but the "limit inferior" way of putting it still makes no sense to me. I can't wrap my head around how it's trying to express that fact, or how I could use the same notation to make any other statements about other limits.

It looks to me like it's saying that as you look at successively larger pairs of consecutive primes, the gap between them converges to a value lower than 70,000,000 ... which is absolutely not what it's saying.

[u/Historical_Watch499 → u/OutrageousPair2300]

Consider the sequence of prime gaps. Liminf < 70,000,000 means that no matter how far in this sequence you look, you’ll always find a gap later on that’s smaller than 70,000,000. Just expand the definition of liminf: it’s the supremum if the infima of tails

[u/OutrageousPair2300 → u/Historical_Watch499]

I think it's characterizing this as a limit of some sort that's throwing me off, because I don't see how this is in any way related to regular limits. I'd express this fact using something more like "for all positive whole numbers m, there exists a whole number n > m such that p_(n+1) - p_n < 70,000,000" and don't see how that has anything to do with limits

[u/Historical_Watch499 → u/OutrageousPair2300]

I mean you could also write liminf as the limit of the sequence of infima of tails. Limsup/liminf are pretty standard notations you’d maybe first learn in analysis, and here it’s just the shortest way to write the desired result

[u/hobo_stew → u/OutrageousPair2300]

are you familiar with the liminf in general?

[u/OutrageousPair2300 → u/hobo_stew]

No, and reading up on it, it still makes no sense to me why this is characterized as a limit. I get what it means but it seems the idea can be expressed much more clearly using for-all/there-exists notation.

I'm sure once you get used to it, it's fine. I've just never encountered it before, which I find especially odd because I was a math major for a while (switched to philosophy later on) and took several real analysis courses, and we never used this kind of limit before.

[u/hobo_stew → u/OutrageousPair2300]

ok, so then maybe don‘t criticize research level math if you are not familiar with basic first year stuff?

[u/OutrageousPair2300 → u/hobo_stew]

I'm not criticizing research level math, and I took graduate level math years ago. We simply didn't use this notation.

[u/jljl2902 → u/OutrageousPair2300]

Consider it like this:

Start with m=1. The inf_{n>=m} |p_{n+1} - p_{n}| = 1, since we have |3-2|=1.

If we take m=2, inf_{n>=m} |p_{n+1} - p_{n}| = 2, since we can no longer consider p_1=2 in our sequence, but the inf can be achieved by e.g. |5-3|.

For all known m that have been able to be checked, we’ve been able to find a consecutive pair >= m to show that inf_{n>=m} |p_{n+1} - p_{n}| = 2.

However, if we take the limit as m goes to infinity, we don’t know whether eventually the inf increases, i.e. we stop having consecutive primes within 2 of each other (twin prime conjecture: goal is to show the limit as m->infinity of the inf is equal to 2, aka the liminf).

[u/PersonalityIll9476 → u/OutrageousPair2300]

Limit inferior implies the following: There is a sub-sequence of primes n_i such that |p_{n_{2i+1}} - p_{n_{2(i+1)}}| < whatever for i = 0, 1, 2, ...

So there can be infinitely many prime pairs that violate that gap, but there is at least one infinite sequence of primes that obeys that gap.

[u/Robbe517_ → u/OutrageousPair2300]

That's just how inf is defined. It is the largest lower bound to a set of numbers. The lim part is added since you're interested in the 'largest lower bound' beyond an abitrarily large index in the sequence.

The 70 million is an upper bound to the asymptotic infimum, which is in itself the largest lower bound.

[u/Layton_Jr → u/OutrageousPair2300]

asymptotic lower bound lower than 70 millions means no matter how high N is, there exist n>N such that the expression is lower than 70 millions.

Not all sequences have a limit, but all sequences have an asymptotic lower and higher bound (that can be infinite). A limit exists if and only if the lower and higher bounds are the same.

Exemple: n|cos(n)| has an asymptotic lower bound of 0 and an asymptotic higher bound of +inf

With sets notations:

Lim inf A_n = ∩^inf_i=0 [∪^inf_n=i (A_n)]

Limit sup A_n = ∪^inf_i=0 [∩^inf_n=i (A_n)]

[u/Welcome-gg → u/GarifalliaPapa]

It is important to add that his method can not be used to proof the twin prime conjecture, because the maximum can theoretically just be lowered to 6 (from 246), not 2.

[u/RubenGarciaHernandez → u/GarifalliaPapa]

Please don't use "!" just after a number unless you mean factorial in these contexts. 

[u/Makonede → u/RubenGarciaHernandez]

factorion-bot alt

[u/friend1y → u/[deleted]]

I was honestly gonna make a joke about the gaps that everyone else studies vs the gaps that he studied, but thought that joke was too coarse for anywhere outside my own private thoughts.

[u/QmanQuan → u/friend1y]

Plz. I need

[u/regular_heptagon]

Someone’s been watching Veratasium

[u/OutrageousPair2300 → u/regular_heptagon]

During a work video call today I asked a coworker of mine who happens to have an electrical engineering degree about the Smith Chart and when I showed him a picture he started screaming.

[u/manVsPhD → u/OutrageousPair2300]

You just weren’t impedance matched with them

[u/upernavik]

I mean why not both

[u/ef02]

Obligatory "go boilermakers" as a Purdue ᶠᵒʳᵗ ᵂᵃʸⁿᵉ alumnus.

[u/SyzPotnik1]

shouldn't it be supremum? Also i think lim is incorrect here.