Is 1 Prime, and Does It Matter?

Why 1 Is Excluded from Primes

• Main practical reason: including 1 would clutter almost every theorem with “prime greater than 1” and “excluding the trivial factor 1,” making exposition worse with little gain.
• The Fundamental Theorem of Arithmetic is cleaner: “every integer > 1 has a unique factorization into primes” vs needing to say “primes > 1” or allow infinitely many 1s in factorizations.
• Related: viewing 1 as the empty product of primes makes it naturally special and non‑prime.
• Several results (Euclid’s lemma, Euler product for the zeta function, sieve of Eratosthenes) behave nicely only if 1 is not prime; treating 1 as prime breaks or trivializes them.

Foundational and Logical Considerations

• Long subthread on how precisely one can “define the natural numbers”:
  • First‑order Peano arithmetic admits non‑standard models (elements behaving like “infinite integers”).
  • Second‑order Peano arithmetic is categorical in full semantics, but then interpretation depends on a richer meta‑theory, leading to regress.
• These issues are used to illustrate that even seemingly simple objects (like ℕ) are subtle to pin down formally, so debates about “is 1 prime” are partly about what framework you adopt.

Empty Set, Zero, and Other Conventions

• Analogy: we could have lived in a mathematical culture that declared the empty set “not a set” to avoid endless “nonempty” qualifiers, but that would complicate other areas (set theory, topology).
• Discussion on whether 0 is a natural number, and parallel notational compromises (ℕ₀ vs ℕ₁, ℤ≥0 vs ℤ>0).
• Some note that including the empty set and 0 usually simplifies algebraic structures (monoids, topologies), though it introduces a few edge cases elsewhere.

Definitions of “Prime” and Generalizations

• Common modern definition cited: a natural number with exactly two (positive) divisors; this neatly excludes 1 and negatives.
• Others point out in algebraic contexts:
  • Distinction between units (invertible elements like ±1) and irreducibles.
  • In other rings (e.g., Gaussian integers), usual primes like 2 may factor further.
• Several commenters stress that prime definitions that exclude 1 are the ones that made unique factorization and algebraic number theory work smoothly.

Programming and Practical Analogies

• Comparison to 0‑based vs 1‑based array indexing: both are conventions, but 0‑based tends to simplify many formulas, just as “1 is not prime” simplifies number theory.
• Example code: defining a product function that returns 1 on an empty list mirrors the empty product convention.
• Some use 0 or −1 as sentinel indices, analogous to how 1 is treated as a “special” but not prime element.

Meta: Axioms, Usefulness, and Philosophy

• Several comments emphasize: axioms and definitions are not “true” or “false,” only more or less useful.
• From this standpoint, the question isn’t “is 1 really prime?” but “does calling 1 prime help?”—consensus in the thread is that it overwhelmingly does not.
• There’s broader reflection on how many philosophical debates (including about infinity, excluded middle, etc.) are ultimately fights over which definitions are most productive.