37, the median value for the second prime factor of an integer

Original Article ↗ Hacker News Discussion ↗

A math article claims that 37 is the “median second prime factor” of a natural number, meaning that if you look at all integers and record, for each, its second-smallest distinct prime factor, then in the long run half of those values are ≤37 and half are ≥37. Commenters unpack what “median” and “second prime factor” mean in this asymptotic sense, why 37 rather than a smaller prime emerges, and how the proof uses density arguments over increasingly large ranges of integers. The thread branches into what makes a number “interesting,” related curiosities about 37 and other integers, and reflections on why such seemingly esoteric facts can still be mathematically satisfying even without practical applications.

Understanding the 37 result

  • Several commenters initially misread the claim, thinking it was “the second prime factor is 37 for half of all numbers,” then corrected to: the probability that the second (distinct, ordered) prime factor is ≤ 37 tends to 50% as numbers grow.
  • Clarified that “median” means half the integers have second prime factor ≤ 37 and half ≥ 37 (in the limiting sense), not that 37 is the most common second factor.
  • A technical explanation from the linked paper’s framework is referenced: asymptotic densities λ₂(p) give the proportion of integers whose second prime factor is p, allowing a limiting median to be identified at 37.

Debates about the median and definitions

  • Some argue the truly interesting part is that a finite median exists at all; once that’s known, it must be some prime.
  • Others question whether it’s legitimate to call 37 the median when cumulative probabilities around 37 are only approximately 50%.
  • There is back-and-forth on:
    • Whether second prime factors can repeat (e.g., for 12: is the second prime 2 or 3?).
    • How to treat 1 and primes themselves (often modeled via an “∞” second factor).
    • How medians are defined for finite sets vs. the N → ∞ limit.
  • A similar issue arises for the first prime factor: depending on conventions for 1 and even/odd N, the limit of medians may fail to exist.

Is 37 really “interesting”?

  • Some embrace 37 as “somewhat interesting,” especially because:
    • It is also the first irregular prime.
    • It shows up in other curiosities (perceived randomness, “move 37” in Go, pop culture references).
  • Others insist 37’s specific value is arbitrary; the structural fact (existence of a finite median) is what’s mathematically notable.

Favorite integers and “most interesting number”

  • Long subthreads debate which integer is “most interesting”:
    • 2, 0, 1 as foundational.
    • 12, 27, 60, 120 for factorization and historical/base-system reasons.
    • Jokes and philosophy about the “interesting number paradox” and why no least-uninteresting integer can exist.
  • Some view primes as less “characterful” than composites; others highlight special primes like 2 and 37.

Practicality vs. pure curiosity

  • One commenter asks what this is “good for” (e.g., cracking lotteries or RSA); most replies frame the result as pure mathematical play, not practical cryptanalysis.
  • Several argue that the interest lies in intuition being overturned: it’s surprising that the second prime factor’s median is so small and can be rigorously pinned down.