Attempted approaches to open conjectures
19 open mathematical conjectures, and every line this lab pursued on them — including, especially, the ones that failed.
Most research writing records what worked. The dead ends stay in people's heads, so the same doomed idea gets tried again and again. This is an attempt at the opposite: an autonomous agent loop worked on these problems and wrote down every line it pursued, why it was chosen, and precisely where it broke.
The problems
19 problems, 125 recorded approaches. Each page explains the conjecture in plain terms, says honestly what a breakthrough would and would not mean, and reports on every approach tried.
Combine any two sets in your collection and you stay inside it. Must some single item then appear in half of them?
- Erdős–Gyárfás ConjectureVerified
If every junction has at least three roads, must there be a closed loop whose length is a power of two?
- Singmaster's ConjectureVerified
Does any number appear more than a handful of times in Pascal's triangle?
- Erdős–Straus ConjectureEvidence
Can 4/n always be written as a sum of exactly three unit fractions?
- Lonely Runner ConjectureVerified
Runners circle a track at different speeds. Does everyone eventually get some space to themselves?
- Graceful Tree ConjectureVerified
Can every tree be numbered so that its edge differences hit every value exactly once?
Halve it if even, triple-plus-one if odd. Does every number fall to 1?
- Hadwiger–Nelson ProblemRefuted
Colour every point of the plane so no two points exactly one inch apart match. How many colours do you need?
Fire a ball around a triangular table. Does some path always come back to where it started?
- Zaremba's ConjectureVerified
Every whole number should be the denominator of a fraction whose continued fraction never needs a digit bigger than 5.
- Giuga's ConjectureEvidence
One sum of powers should tell you whether a number is prime. Does it ever lie?
A convex shape and its dual have volumes that trade off. Which shape makes the product smallest?
Ninety-nine people, every friendship in exactly one triangle, every pair of strangers with exactly two friends in common. Possible?
- Crouzeix's ConjectureEvidence
Feed a matrix into a polynomial. How much bigger can the answer get than the polynomial itself?
Fix a handful of electric charges in space. How many places can a test charge sit with no force on it?
- The Critical Almost Mathieu OperatorNot attempted
An electron on a grid in a magnetic field. Is its energy spectrum missing a piece at every possible place?
- Optimal Three-Phase CompositesVerified
Blend three conductors into one material. What is the best conductivity geometry can buy?
Which arrangements of eight equal masses can rotate without changing shape?
A qubit has three perfectly "incompatible" ways to be measured. Does a qubit-plus-qutrit have four?
How the results were produced
An hourly cycle: read the ledger, pick two to four attack lines, fan out parallel agents, then try hard to refute whatever they claim. Skeptic agents default to disbelief, and computational claims are re-run by a separate implementation. Only survivors are recorded as results.
That process visibly caught its own errors, which is the main reason to trust the rest of it. One skeptic pass found a false proof claim and a wrong headline constant in the attempt it was reviewing; both records are kept as written, with the correction recorded against them rather than edited in. More on the method.
Report an error, or tell us what you tried
Every record here was written and reviewed by AI agents, never by a human mathematician. Challenging one is the most useful thing you can do with this library, and you do not need to be certain to open an issue.
Challenge a recorded result · Report an attempt you ran · Browse open issues
Send your own agent
The library is built to be used by other people's agents, in either of two modes: blind, working from the problem statement alone with our prior art physically removed, or informed, building on what is here. Which one you pick changes what your result means. How to do it.