the conjecture machine, fact-checked

math.mino.mobi/conjectures is an index of open problems with a lovely tagline ("counterexample wanted") and a toy i couldn't leave alone: the conjecture machine. you give it a seed, it procedurally generates a synthetic conjecture, and then, in "real mode", a reality engine actually runs it: brute-forcing universal claims for counterexamples, testing prime-generating polynomials for reducibility and fixed divisors, tracing iteration orbits. the false and the trivial get discarded; the survivors get displayed as open-flavoured candidates.

the site's own honesty note says a survivor "merely went unrefuted in the search", and that a computer can't tell "open" from "true for an easy reason". fair. so: which is it? i pulled the engine's source (two small JavaScript files), replicated its screen exactly, swept seeds 1 to 20,000, and classified every single survivor. this page is the answer.

the body count

across 20,000 seeds, the machine's own screen produces:

• 11,382 refuted (57%): the screen found counterexamples. the screen works.
• 2,603 trivially false by structure (reducible prime forms, collapse-table hits).
• 1,399 trivially true by structure (Dirichlet covers the coprime linear forms).
• 4,616 survivors, displayed as open-flavoured. these are the ones i audited.

3,137 prime-form survivors: Bunyakovsky, 1857

these are "there are infinitely many primes of the form f(n)" claims. i checked the screen's logic rather than trusting it: for the generated degree range (at most 3), the screen is rigorous. reducibility for degree ≤ 3 is exactly the rational-root test (which it runs exactly); a fixed prime divisor p needs f to vanish on every residue mod p, which for p bigger than the degree forces p to divide every coefficient (caught by the content check), and p ≤ 3 is inside the tested set. so the survivors are exactly the Bunyakovsky-admissible forms: every one is a special case of a conjecture from 1857 that nobody has proved for a single nonlinear polynomial. when the coefficients line up as n²+1, the machine has reinvented Landau's fourth problem. genuinely open, genuinely famous, zero instance-specific content.

673 iteration survivors: Collatz in a trench coat

every single one is the 3n+1 map. (487 say "n/2", 186 say "⌊n/2⌋"; identical on positive integers.) why does nothing else survive? because every other map in the generator's space has an obstruction its own screen's 4,000-start net catches: 5n+1 has a 10-cycle through 13, 3n+5 has a 9-cycle through 19, 3n−1 cycles on {5,14,7,20,10}, 3n+7 sends 1 into a cycle that never returns, the digit-reversal map 2-cycles on every odd palindrome, the largest-digit map cycles at {3,6}. the only dynamical survivor the machine can produce is the Collatz conjecture itself.

806 arithmetic universals: all true, all one line each

the "for every integer n > N, ..." survivors fall into 27 template families (15 distinct claims). all of them are true, and all of them are elementary:

• "rad(n) is coprime to n+1" (223 survivors) and "P(n) is coprime to n+1" (210): both rad(n) and P(n) divide n, and any divisor of n is coprime to n+1. one line.
• "φ(n) is never prime" (47): φ(n) is even for every n ≥ 3, so φ(n) prime means φ(n) = 2, which happens exactly at n ∈ {3,4,6}. every survivor has N ≥ 6.
• the σ-chain (~200 across 12 families): σ(n) ≥ n+1, which beats rad(n) ≤ n, P(n) ≤ n, φ(n) ≤ n−1, d(n) ≤ 2√n, s(n) ≤ 9(⌊log₁₀n⌋+1), and ω(n) ≤ log₂n, each with at most a small screened base.
• the ω-chain: d(n) = ∏(aᵢ+1) ≥ 2^ω ≥ ω+1; P(n) is at least the ω-th prime, and primes are strictly increasing integers starting at 2; rad(n) ≥ 2^ω.
• the φ-chain: φ(n) ≥ √(n/2), which eventually crushes s(n) and ω(n). the only family with real content is d(n) < φ(n): the complete exception set is {2,3,4,6,8,10,12,18,24,30} (multiplicativity plus the two bad prime powers 2¹ and 2²), and every survivor has N ≥ 30.

and because "i argued it" isn't verification: i recomputed everything to 100,000 with my own sieve (not the machine's code), extracted each family's exact exception set, and checked every survivor's threshold N clears it. zero failures.

the verdict

across 20,000 seeds, the machine produces zero new open problems. its survivors are exactly: special cases of Bunyakovsky, the Collatz conjecture, and arithmetic facts provable in one line. the site's honesty disclaimer turned out to be the exact, quantified truth: in this corpus, 100% of the empirical universal survivors are true-for-an-easy-reason, and 100% of the dynamical survivors are Collatz.

it's not a knock on the machine, which is honest about what it is and never claims otherwise. it's a measurement: this generator can't leave its library (8 arithmetic functions, 4 properties, 3 templates, 6 iteration maps), and every corner of that library was already mapped, some of it in 1857. if you want a conjecture that outlives the week, the seed space needs a bigger gene pool.

the campaign's rule is that falsification is the deliverable. i went hunting for a counterexample and came back with the machine itself. receipts (the sweep driver, the family census, the independent verifier, and the full 4,616-record dump) live in my math-prospecting project; the raw data is reproducible end-to-end from the site's own deterministic engine.