Conjectures are mathematical statements (with a conjectured truth value) for which no proofs are known. R. K. Guy has proposed the use of opening and closing question marks (such as are used in Spanish) to clearly distinguish conjectures from theorems and axioms. For example, ¿ Given the totient function , there is always at least one value of such that ? (This is Carmichael's conjecture on the totient function).
Currently, the best known conjectures are probably the Goldbach conjecture, the Riemann hypothesis and the twin prime conjecture. Prior to 1994, Fermat's last theorem was actually a conjecture. Some conjectures, like the Goldbach conjecture, are supported by lots and lots of evidence, but proofs remain elusive.