Checking a trillion cases still does not prove a conjecture
A mathematical conjecture is a proposition offered without proof. The Collatz conjecture has been tested past 1.2 trillion without a counterexample, yet one single exception would destroy it. Fermat's conjecture took 358 years; the four color theorem needed a computer checking 1,936 maps.
In mathematics, a conjecture is a tentative proposition lacking proof. Formal mathematics demands provable truth: no amount of confirming examples proves a universal claim, because one counterexample suffices. Mathematicians may still treat well-tested conjectures as strongly supported when consequences align with known results.
The four color problem itself began on 23 October 1852, when Francis Guthrie noticed that four colours sufficed for a map of English counties. Proof methods vary. Brute force checks every finite case—used for the four color theorem's 1,936 unavoidable maps in 1976 by Kenneth Appel and Wolfgang Haken, the first major computer-assisted proof. Doubts about machine verification lingered until 2005 confirmation by theorem-proving software. Once proved, a conjecture becomes a theorem.
Fermat's Last Theorem, conjectured by Pierre de Fermat in 1637, was proved by Andrew Wiles in 1994 and published in 1995 after 358 years. It once held a Guinness record for difficulty. The Poincaré conjecture, posed by Henri Poincaré in 1904, was proved by Grigori Perelman in 2002–2003 using Ricci flow with surgery.
Not every conjecture resolves true or false. The continuum hypothesis is independent of Zermelo–Fraenkel set theory—you may adopt it or its negation consistently. The Riemann hypothesis, proposed by Bernhard Riemann in 1859, remains among pure mathematics' most important open problems and is a Clay Millennium Prize question. Mathematicians sometimes build conditional proofs that assume such conjectures in their hypotheses until verification arrives. The Hauptvermutung of topology, proposed in 1908, was later shown false by John Milnor in 1961. Its manifold version does hold in low dimensions, settled by Tibor Radó in the 1920s and Edwin Moise in the 1950s. Other guesses collapse to a single example, a fate shared by Euler's conjecture on sums of powers and by the Pólya conjecture.
Source: Conjecture
