from Hacker News

Mathematicians hunting prime numbers discover infinite new pattern

by georgecmu on 6/19/25, 3:28 AM with 101 comments

  • by munichpavel on 6/21/25, 4:33 PM

    Ken Ono, one of the authors, is the mathematician behind the University of Virginia women's swimming team's dominance in recent years, including world records and gold medals.

    https://news.virginia.edu/content/faculty-spotlight-math-pro...

  • by gnabgib on 6/21/25, 5:19 AM

  • by wewewedxfgdf on 6/21/25, 4:48 AM

    This sort of thing makes me feel there is some deep understanding of reality only inches away from us, we glimpse it through these patterns but the secret remains hidden.

  • by Sniffnoy on 6/21/25, 5:38 AM

    I'm a little confused at the significance here. Before I read the definition of the M_a, this seemed crazy, but on actually reading it, M_1 is just the sum-of-divisors function (usually denoted sigma).

    So, n is prime iff M_1(n)=n+1. That's much simpler than the first equation listed there!

    Indeed, looking things up, it seems that in general the functions M_a can be written as a linear combination (note: with polynomial coefficients, not constant) of the sigma_k (sigma_k is the sum of the k'th power of the divisors). So this result becomes a lot less surprising once you know that...

  • by ysofunny on 6/21/25, 4:23 PM

    I hope the twin prime conjecture will become a theorem during the remainder of my lifetime

    that's why I already got the double twin prime conjecture ready:

    there exists an infinite number of consecutive twin primes. 3 examples: 11,13; 17,19. 101,103;107,109, AND 191,193;197,199... I know of another example near the 800s

    there's also the dubious, or trivial, or dunno (gotta generalize this pattern as well) of the first "consecutive" twin prime but they overlap which is 3,5 and 5,7.... which reminds me of how only 2 and 3 are both primes off by one; again, I need to generalize this pattern of "last time ever primes did that"

  • by noqc on 6/21/25, 4:36 PM

    Because the article doesn't actually say so (presumably because the author doesn't know the difference between "if" and "if and only if") the statement:

    (3n^3 − 13n^2 + 18n − 8)M_1(n) + (12n^2 − 120n + 212)M_2(n) − 960M_3(n) = 0

    is equivalent to the statement that n is prime. The result is that there are infinitely many such characterizing equations.

  • by andsoitis on 6/21/25, 5:12 AM

  • by anthk on 6/21/25, 12:37 PM

    Prime generating functions in polynomials? That's almost Lisp domain.

    Mathematicians should play with Scheme and SICP.

  • by coderatlarge on 6/24/25, 10:21 AM

    just this quote from the article shocked me:

    “In 1976, Jones, Sato, Wada, and Wiens (2) (…) produced a degree 25 polynomial in 26 variables whose positive values, as the variables vary over nonnegative integers, is the set of primes.“

  • by drdunce on 6/21/25, 4:18 PM

    This have implications for public key cryptography?

  • by nprateem on 6/21/25, 3:21 PM

    Oh. (3n3 − 13n2 + 18n − 8)M1(n) + (12n2 − 120n + 212)M2(n) − 960M3(n) = 0.

    I'd have thought that was obvious.

  • by swayvil on 6/21/25, 1:14 PM

    Yeah but can we get a pretty picture out of it? A cool fractal is worth a thousand words.