Conjecture
A conjecture is a mathematical statement believed to be true based on evidence or intuition but not yet rigorously proven.
Definition
A conjecture is an educated guess in mathematics: you think it is true based on patterns you have seen, but you have not proven it yet. More precisely, a conjecture is a proposition believed true based on incomplete evidence or pattern recognition, but lacking formal proof, supported by heuristic evidence, special-case verification, or probabilistic arguments; disproving it requires only a single counterexample. In number theory, analytic methods and computational verification can confirm conjectures for enormous ranges without constituting a proof, and some conjectures turn out to be independent of ZFC, meaning neither they nor their negations can be proven.
Example
Noticing $2+2=4$, $4+4=8$, $6+6=12$, you might conjecture "every even number is the sum of two primes," which is Goldbach's Conjecture, still unproven! Fermat's Last Theorem was a conjecture for $358$ years, that no integers $a, b, c$ satisfy $a^n + b^n = c^n$ for $n > 2$, until Andrew Wiles proved it in 1995, transforming it into a theorem. The Riemann Hypothesis, that all non-trivial zeros of the Riemann zeta function have real part $1/2$, has been verified for the first $10^{13}$ zeros but remains unproven; a proof would determine the distribution of primes with unprecedented precision.
Key Insight
A conjecture becomes a theorem once it is proven; until then, even the most obvious-seeming patterns need proof. Conjectures drive mathematical research: the Riemann Hypothesis and Collatz Conjecture remain unproven despite enormous effort, suggesting these seemingly simple patterns hide profound depth, a possibility that must always be considered when a statement resists proof.