Alphabeta Math

Primes and Factorisation: Examples and Counterexamples

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How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

✓ 9 results · all verified · 3 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full by a delegated reviewing agent on the owner's instruction; the judge is an additional, independent cross-model AI review of the proofs. The 6 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Combinatorics
Number Theory
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

360 = 2³ · 3² · 5 and 84 = 2² · 3 · 7, with gcd(360,84) = 12 and lcm(360,84) = 2520 read off the exponents

Example

2 is prime, and it is the only even prime: every even integer n > 2 is composite

Example

No rational squares to 3 or to 6, and none cubes to 2: three instances of the rational-root corollary

Example

For every n ∈ ℕ there are n consecutive composite integers: with N := ∏_j<n(j+2), each of N+2, …, N+n+1 is composite

Counterexample

In the multiplicative monoid H = {1, 4, 7, 10, …} of positive integers one more than a multiple of 3, the element 100 has two genuinely different factorisations into irreducibles, 4 · 25 and 10 · 10

Counterexample

If 1 were admitted as a prime, uniqueness would fail: 6 = 2 · 3 = 1 · 2 · 3 = 1 · 1 · 2 · 3, lists of different lengths that no permutation matches

False statement

FALSE: for every finite list p₀, …, pₙ₋₁ of distinct primes, p₀ ⋯ pₙ₋₁ + 1 is prime

False statement

FALSE: n² + n + 41 is prime for every natural number n

False statement

FALSE: every Fermat number 2^2ⁿ + 1 is prime

Sources