Mathematicians sometimes sense a pattern before they can prove it. A conjecture is a statement that seems true based on evidence, but lacks a rigorous proof. It sits in the gap between observation and certainty.

Goldbach's conjecture, proposed in 1742, says every even integer greater than 2 is the sum of two primes. It has been checked for enormous numbers, yet no proof exists. The Collatz conjecture, about a simple iterative process, remains open despite decades of effort. Fermat's Last Theorem was a conjecture for over 350 years until Andrew Wiles proved it in 1994.

Conjectures drive research. They focus attention, suggest new methods, and sometimes turn out to be false. A counterexample can destroy a conjecture overnight. The pursuit of proof, not the conjecture itself, often produces the deepest mathematics.

• Statement believed true but unproven
• Goldbach and Collatz are famous examples
• Can be disproved by a single counterexample
• Drive research and new methods