A conjecture is something that a mathematician thinks is probably true, but has not been able to prove nor disprove. There are some widely-known conjectures, b

A conjecture is something that a mathematician thinks is probably true, but has not been able to prove nor disprove. There are some widely-known conjectures, but most are just something a mathematician ran into while thinking about some other problem and then ran into a dead end while trying to prove/disprove it. Often, a conjecture will be invented and then proved or disproved a short time later (I remember my wife disproving one her thesis adviser had proposed in class earlier that day), but occasionally one will remain unproven either way for a long time. With many of the "big" conjectures, this gives people confidence they're true -- they've been tested for millions, billions, or even trillions of cases over the course of decades or centuries, but nobody has found a counterexample. But they also haven't come up with an airtight proof; maybe they'd fail on the ten trillionth try, or maybe they'd hold true forever. That uncertainty is part of what makes them interesting.