Mathematicians hunting prime numbers discover infinite new pattern

Big-picture reactions: primes, patterns, and “ultimate reality”

• Several comments frame the result as a tantalizing “glimpse” of some deep structure, akin to Plato’s cave or the Mandelbrot set.
• Others push back: they see this more as exploring the structure of discrete math, not the structure of physical reality itself.
• There’s also the classic “it’ll turn out to be trivial in hindsight” sentiment, contrasted with the possibility that maybe there is no deep pattern to primes at all—and both paths are seen as still worthwhile for the journey.

Math vs reality and discreteness

• Debate over whether discrete math is the most “observed property of reality” or purely an abstraction layered on top of continuous or unified phenomena.
• Examples with apples, rabbits, and virtual objects illustrate that “2” depends on classification and cognitive abstraction.
• Discussion touches on whether spacetime is discrete (Planck units) vs a continuous manifold, and the possibility that space and time are emergent rather than fundamental.
• General theme: counting and measurement are powerful but psychologically-loaded abstractions.

Primality testing and cryptography relevance

• Some wonder if a “simple way to determine primeness without factoring” might exist and be overlooked.
• Primality tests that don’t require factoring are noted (e.g., Lucas–Lehmer for Mersenne numbers, probabilistic tests, AKS), with the observation that these have been known for decades.
• On cryptography: commenters think this specific result is unlikely to matter, since computing the involved functions (e.g., M₁) seems at least as hard as factoring.

Significance and technical content of the new result

• The article’s central equation is noted to be an “if and only if” characterization of primes; the paper proves there are infinitely many such characterizing equations built from MacMahon partition functions.
• One line of discussion: M₁ is just the sum-of-divisors function σ(n), so the trivial characterization “n is prime ⇔ σ(n) = n+1” already exists; this makes the new formulas feel less astonishing.
• Others reply that the novelty lies in:
  • Connecting MacMahon’s partition functions to divisor sums in a nontrivial way.
  • Showing specific polynomial relations of these series that detect primality.
  • A conjecture that there are exactly five such relations, which is seen as “spooky” and suggestive of deeper structure.
• There is a side debate on the meaning of “iff,” with clarifications that “A iff B” means mutual logical implication, not uniqueness of representation.

Related curiosities and generalizations

• Mention of highly complicated prime-generating polynomials (e.g., Jones–Sato–Wada–Wiens) as a conceptual parallel.
• Brief discussion of twin primes, “consecutive twin primes,” and their generalization to broader conjectures (Dickson, Schinzel’s Hypothesis H).