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 ... 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

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

... 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 ... "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.

@#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.

@#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.