conjectures — an index of open mathematical conjectures

Every entry below is open — nobody knows whether it is true or false — and almost all of them share a shape: they assert that something holds for all objects of some kind. That makes them fragile in a specific, beautiful way. You don't need a hundred-page proof to settle one. You need a single counterexample: one even number that isn't a sum of two primes, one graph that can't be reconstructed from its pieces, one starting value that sends the Collatz map to infinity. Find it and the conjecture falls.

This page is the index — and, by default, a difficulty ranking. Every entry carries a score: our estimate of the probability it is still unproven in 2126, a hundred years from now. High score = deep, impervious, no line of attack (Collatz, the Riemann hypothesis); low score = a case is falling or a proof is already circulating (the moving-sofa problem, the lonely runner). The table opens sorted hardest-first; the # column is each conjecture's rank across the whole list. Sort by any other column, filter by field, or search — then click a row for the full context page.

The score is a considered editorial estimate, not an oracle — it weighs depth, how long the problem has already resisted, and current momentum. Calibrated against fixed anchors so the ranking is consistent across fields; treat it as a conversation-starter, not a betting line.

the conjecture machine · can a machine make new ones? Spin the generator — it invents synthetic conjectures from a seed, then a reality engine actually runs each one: brute-forcing ∀-claims for a counterexample, testing prime-form polynomials for reducibility and fixed divisors, tracing iteration orbits for divergence. It discards the false and trivial and surfaces only survivors — occasionally reinventing a real open problem. Deterministic, permalinkable, and honest about what "survived" means.

How to read the disproof column

On each context page a counterexample tag means a single explicit object refutes the conjecture — the classic hunt. counterexample (deep) means a counter-object exists in principle but exhibiting or certifying it is itself hard (a Riemann zero off the line, a Navier–Stokes solution that blows up). existence marks the mirror-image problems — they claim something exists, so the counterexample would be a proof that it never does. We keep those because they're the natural companions to the hunt, but they don't yield to a lucky search the same way.