Breakthrough a step toward revealing hidden structure of prime numbers

This is from May and there was a better article in Quanta already discussed here.<p><a href="https:&#x2F;&#x2F;www.quantamagazine.org&#x2F;sensational-proof-delivers-new-insights-into-prime-numbers-20240715&#x2F;" rel="nofollow">https:&#x2F;&#x2F;www.quantamagazine.org&#x2F;sensational-proof-delivers-ne...</a>

This got me thinking.<p>Imagine this discovery led to a larger breakthrough on prime numbers that allowed easy factorization of large integers and effectively rendered public key cryptography such as RSA ineffective overnight, by allowing anyone with a consumer-grade CPU to crack any production-size key.<p>Does the industry have DR plans for this scenario? Can the big players quickly switch to a different, unbroken encryption system? While it would probably be a heavenly day for jailbreakers, console modders and other &quot;device freedom&quot; types generally, the overall impact would be disastrous and incalculable.<p>Does the industry simply not consider &quot;sudden number theory breakthrough&quot; a possible event?

People always think the structure of primes is complex, but it&#x27;s not really, it&#x27;s just a recursive structure of the magnitude gaps not landed on by multiples of previous gaps.<p>It doesn&#x27;t make it easier to &quot;predict&quot; without tracking all prior gaps, but it&#x27;s not essentially a complex structure. Kind of funny that like such a simple structure is so elusive. Sorta like how the 3n + 1 sequence gives rise to such complexity. Or the logistic map with its parameter above the threshold.

Something inspiring about this: &quot;In dedicated Friday afternoon thinking sessions, he returned to the problem again and again over the past decade, to no avail.&quot;

If you plot the Gauss and Riemann curves in a specific space you see something more magical....<p>To see what I am talking about as in trivial and non-trivial zeros see this wikipedia animation <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;File:Riemann3d_Re_0.1_to_0.9_Im_1_to_51.ogg" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;File:Riemann3d_Re_0.1_to_0.9_I...</a><p>Basically, it implies that there is another relationship between real and imaginary numbers we have not yet stumbled upon....<p>And,this has implications upon finding the gravity theory as Riemann math is involved in quantum mechanics....<p>Strange science that primes is or might be involved in gravity theory....

&quot;they pulled some unorthodox moves to finally break Ingham’s bound&quot;<p>Why is taking methods from other fields an unorthodox move? I come from an engineering background an there it is the common case. The usage of harmonic analysis is a staple in many fields (audio, waves, electrical analysis, statistics) and of course the algorithms are pure math under the hood. If I want to find a reaccuring structure in an underlying system, wouldn&#x27;t it be normal to try different plotting techniques and choose the one that suits my problem best?

&gt; “At first sight, they look pretty random,” says James Maynard, a mathematician at the University of Oxford. “But actually, there’s believed to be this hidden structure within the prime numbers.”<p>What would the pattern of primes hypothetically look like? Is there expected to be some kind of closed form formula? If the Riemann hypothesis were proven, what would be the next step to understanding the distribution? Or is the proof itself expected to hold this answer?

Every time I hear about James Maynard it really solidifies my opinion that he&#x27;s one of those once in a generation geniuses. He&#x27;s already contributed so much to prime number theory, it really feels like there might be a proof of the Riemann Hypothesis within my lifetime.

I’m curious as I hadn’t seen it before and it’s gripping: Is the patterns showing in a polar plot of the prime numbers a recent discovery or is it long known and just used as an illustration? What is it called and what is its history?

On a slight tangent, this line makes me think about aspects of automated provers that I don’t even know if we’ve begun thinking about:<p>&gt; “It’s a sensational breakthrough,” says Alex Kontorovich, a mathematician at Rutgers University. “There are a bunch of new ideas going into this proof that people are going to be mining for years.”<p>Frequently, a proof of a thing is less interesting as a way to bring rigor than it is as a new way to look at a thing. I wonder if there’s been any work on that side of things in automated mathematics?

I’m both a layman and a simpleton, but seeing Guth’s comments, surely it can’t be a new idea that the fundamental interpretation of primes is something to do with waves and harmonics?

How is this any different from Sach&#x27;s original work from 2003?<p><a href="https:&#x2F;&#x2F;naturalnumbers.org&#x2F;sparticle.html" rel="nofollow">https:&#x2F;&#x2F;naturalnumbers.org&#x2F;sparticle.html</a><p>The organized patterns of primes and composites was an understood feature of the Sack&#x27;s Spiral since the day he published his findings online.

Reminds me of a story where some egghead friend of mine had a friend that was a researcher at a state school in California.<p>In his research, he found something like getting unenriched uranium to react (please excuse my complete lack of familiarity with the subject).<p>Apparently some government agency stepped in, classified his research and asked him to start.<p>Makes me where else this might have happened - there must be some interesting stuff out there.

3 years ago, somebody post on HN, an animation about prime numbers, it was beautiful looking how prime numbers show a pattern, it looks like the image in this article

“This left a small but unsettling possibility that many zeros could be hiding out right at three-quarters.”<p>Ok, but if zeros there are found some mathematicians may as well call them “trivial zeros.” Can there be an objection to that?

Just use 42 everywhere

And if they crack that, well security is pretty much cracked too...

&quot;analyze this for hidden underlying structure or emergent properties&quot;<p><a href="https:&#x2F;&#x2F;chatgpt.com&#x2F;api&#x2F;content&#x2F;file-HFFSXBEAtdR1fbum5ZCEloge" rel="nofollow">https:&#x2F;&#x2F;chatgpt.com&#x2F;api&#x2F;content&#x2F;file-HFFSXBEAtdR1fbum5ZCElog...</a>

I&#x27;ve been fascinated by this question since I learned the sieve of eratosthenes as a kid. The meta logic of it is so simple:<p>Primes are specifically the numbers that are left over after the structured numbers (composite) ones are removed.<p>Everything - [structured numbers] = [ chaos? the abyss? some meta structure? ]