Monday, June 23, 2025

Remarkable Pattern Discovered Behind Prime Numbers

"The good news is that the world will still be safe."

Posted by HeliumRat at 09:32 PM | 27 COMMENTS | permalink

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"The prime numbers [ ... ] are the solutions of infinitely many special Diophantine equations' in well-studied partition functions," they explain in their new paper.

"In other words, integer partitions detect the primes in infinitely many natural ways."

Comments

More from the cited article ...

... In the third century BCE, a clever Greek chap by the name of Eratosthenes came up with a novel new piece of mathematics: a "sieve", by which one could painstakingly trawl through every integer in order and discard any with more than two factors, leaving only the primes behind.

It was smart for the time -- but objectively, you've gotta admit it's pretty elementary. No offence to Eratosthenes, but that's probably exactly the method a schoolchild would use to find and list the primes, too. But here's the thing: more than two millennia later, it's still one of the best methods we have for the task. ...


#1 | Posted by LampLighter at 2025-06-22 09:12 PM

@#1 ... you've gotta admit it's pretty elementary. ...

No, what you've gotta admit is that we have learned a lot since then.

OK, moving on ...

#2 | Posted by LampLighter at 2025-06-22 09:13 PM

@#2

More from the cited article ...

... It's a testament to just how peculiar and pesky the primes are. Finding any kind of sense or pattern to these numbers has, for centuries, been mathematics' white whale: cropping up seemingly randomly in the number line, they evade prediction and categorization, forming a natural blockade against progress in myriad open problems.

At least, until recently. Last year, a trio of mathematicians found what seemed to be a kind of order in the primes -- and it came from somewhere completely unexpected.

"This paper connects two fundamental areas of number theory: prime numbers and partitions," said Ken Ono, Marvin Rosenblum Professor of Mathematics at the University of Virginia and one of the authors behind the new discovery, in a recent statement.

"Although prime numbers have been studied for centuries, many of their most basic properties remain elusive," he said. "What we proved gives infinitely many new ways to detect prime numbers without having to check divisibility, which is one of the reasons primes are so difficult to detect." ...



#3 | Posted by LampLighter at 2025-06-22 09:21 PM

@#3

... and that matters... why?

Maybe this ...

Yet more from the cited article ...

... It's big news " so much so that Ono was named as a runner up for the 2025 Cozzarelli Prize in the physical sciences, which recognizes teams "whose PNAS articles have made outstanding contributions to their field." So it behooves us to ask, really: what's the big deal? ...

"Large prime numbers are used prominently in other cryptosystems too," Weiss, who was not involved in the new research, added.

The basic idea behind all these systems is the same, though: it relies on the fact that finding primes is a very difficult task. After all these years, finding some kind of new insight into them would require a really innovative perspective -" something nobody had tried before.

Luckily, that's exactly what Ono and his colleagues had. ...


#4 | Posted by LampLighter at 2025-06-22 09:23 PM

@$4 ... "Large prime numbers are used prominently in other cryptosystems too ...

Yeah, security systems seem to rely on prime numbers not being able to be factored.

Does this change that?

I wish I knew. I've not yet delved into the paper.

But it has certainly gained my attention.

#5 | Posted by LampLighter at 2025-06-22 09:26 PM

Huh. Let a, b and c be integers with a0 and b0, and let d=gcd(a,b). If d does not divide c, then the linear Diophantine equation ax+by=c has no solution.

I'll have to take your word on that one.

#6 | Posted by HeliumRat at 2025-06-22 09:33 PM

That should read "a != 0, b != 0". gcd means greatest common divisor.

#7 | Posted by HeliumRat at 2025-06-22 09:39 PM

@#6 ... I'll have to take your word on that one. ...

I'm not proffering anything, so no reason to take my word on anything on this thread.

All I am saying is that if it seems to be easier to factor prime numbers, then that will have an effect upon the security algorithms that crypto-security seems to rely on.

Nothing more, nothing less.


#8 | Posted by LampLighter at 2025-06-22 09:46 PM

#8 Not you, "them". As in "they" (but not the gender "they").

I think they are saying because 4=1+1+1+1 and 2+2 and 2+1+1, you only have to look for primes in it's greatest common divisor, like 2?

I'm going to find this paper. A lot of times math like this goes beyond my pay grade but it's still fun to try to understand.

#9 | Posted by HeliumRat at 2025-06-22 10:00 PM

@#9 ... A lot of times math like this goes beyond my pay grade but it's still fun to try to understand. ...

You and me both.

#10 | Posted by LampLighter at 2025-06-22 10:09 PM

Math is hard

#11 | Posted by LegallyYourDead at 2025-06-23 02:44 PM

Yeah, security systems seem to rely on prime numbers not being able to be factored.
Does this change that?
I wish I knew. I've not yet delved into the paper.
But it has certainly gained my attention.
#5 | Posted by LampLighter

Easily factoring multiples of large prime numbers is also under threat from quantum computing. And while there are quantum safe encryption methods that the big names have already advised the public to start using, it seems like we're going to a place where no encryption lasts forever.

#12 | Posted by snoofy at 2025-06-23 10:12 PM

So I looked up how to find if it's a prime number, surely there's a Wikipedia article for that.
Primality Test
en.wikipedia.org

The fastest current test for if a number is prime is order (log n)^6, where n is the very large prime number.

Reading the paper "Integer partitions detect the primes"
https://www.pnas.org/doi/10.1073/pnas.2409417121
They present a cubic equation, aka order n^3.

So whether a cubic equation in n, which is a very large prime, is faster than a sixth power equation in log n, which would be a much smaller number, I really couldn't say, but I wouldn't be surprised if this finds its way into computer science.

#13 | Posted by snoofy at 2025-06-23 10:32 PM

This sounds like the pattern Carl Sagan speculated might exist and if ever found would prove the existence of God.

#14 | Posted by Tor at 2025-06-24 07:01 PM

Meaning can well indicate mind. This discovery follows well after the Human Genome was 'discovered'; at the time the world's longest 'word'.

There's still a debate over whether the maths am invented or discovered.

#15 | Posted by Corky at 2025-06-24 07:28 PM

IMHO you can invent ways to apply math but you can only discover math patterns.

#16 | Posted by Tor at 2025-06-24 07:36 PM

@#14 ... the existence of God. ...

Meaning that all we know about the universe may be little more than a petri dish filled with agar?


#17 | Posted by LampLighter at 2025-06-24 07:41 PM

@17

why does it surprise you we know little about the universe?

It's rather big for us to understand.

#18 | Posted by Tor at 2025-06-24 08:07 PM

How is it that hardly any major religion has looked at science and concluded, "This is better than we thought! The Universe is much bigger than our prophets said, grander, more subtle, more elegant?" Instead they say, "No, no, no! My god is a little god, and I want him to stay that way." A religion, old or new, that stressed the magnificence of the Universe as revealed by modern science might be able to draw forth reserves of reverence and awe hardly tapped by the conventional faiths.

Carl Sagan

#19 | Posted by snoofy at 2025-06-24 08:29 PM

"There's still a debate over whether the maths am invented or discovered."

Babylonians invented the zero, in terms of writing down a thing that means zero, and in terms of having an exponent based system for writing numbers.

Those I think are closer to what we can call inventions.

I feel like in math we occasionally invent tools, say you're Isaac Newton and you invent calculus. That invention provides a new way to solve existing problems, resulting in new discoveries.

#20 | Posted by snoofy at 2025-06-24 08:42 PM

- say you're Isaac Newton and you invent calculus.

Who's greatest interest, and what he wrote about most, was who he thought had created the universe.

#21 | Posted by Corky at 2025-06-24 08:48 PM

He also spent most of his science career trying to turn lead into gold.

Man was a total goober.

But, he was smart enough to stay home during an outbreak of plague. He spent that month calculating pi to some pointlessly high number of decimal places; pointless even by modern NASA standards.

#22 | Posted by snoofy at 2025-06-24 08:50 PM

#22

He would be happy to know that CERN finally did that for him, however briefly.

Of course, the guy who led the Human Genome Project is both a Darwinist and a Godder, so it takes all kinds. Also, the Nobel Prize for Physics has been won by agnostics, atheists, and theists, so everyone is apparently eligible.

One of the strongest advocates for Design in recent years is also an Oxford Prof specializing in in pure mathematics... and who has spoken much about the patterns thereof pointing more towards meaning than chaos.

www.youtube.com

#23 | Posted by Corky at 2025-06-24 09:10 PM

Well... In ninth grade I got a C in geometry... And in college I got a C- in statistics. (Though marijuana use did play a role in that Stats grade.)

#24 | Posted by moder8 at 2025-06-25 02:04 PM

#24

I also got a C- in geometry... and an A in psychological statistics, prolly only because I was interested in the latter.

(though acid may have played a roll in both)

#25 | Posted by Corky at 2025-06-25 03:02 PM

This is pretty interesting:

Is Our Universe Inside a Black Hole? This Makes it Plausible

"Physicists say we live in a black hole. Not metaphorically, not emotionally, and not just during tax season. But literally, inside a black hole. And not just this.

A group of researchers has published a paper claiming that we've got the Big Bang wrong. It was actually the collapse of an earlier universe to a black hole, which we now live in. That sounds ... wild. I've had a look."

www.youtube.com

#26 | Posted by Corky at 2025-06-25 03:30 PM

The universe is kinda big to be inside a black hole.

Besides if we were in one that would mean there are black holes within black holes.

#27 | Posted by Tor at 2025-06-26 01:24 PM

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