Conjecture
Calculus & Advanced MathA 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.
Example
You might notice $2+2=4$, $4+4=8$, $6+6=12$ and conjecture "every even number is the sum of two primes." That is actually Goldbach's Conjecture, still unproven!
Key Insight
A conjecture becomes a theorem once it is proven. Until then, even the most obvious-seeming patterns need proof.
Definition
A conjecture is a proposition believed true based on incomplete evidence or pattern recognition, but lacking formal proof. Famous conjectures include Goldbach's (every even integer $> 2$ is a sum of two primes), proved computationally for huge numbers but never generally proven.
Example
Fermat's Last Theorem was a conjecture for $358$ years: no integers $a, b, c$ satisfy $a^n + b^n = c^n$ for $n > 2$. Andrew Wiles proved it in 1995, transforming it into a theorem.
Key Insight
Conjectures drive mathematical research. The Riemann Hypothesis and Collatz Conjecture remain unproven despite enormous effort, suggesting these seemingly simple patterns hide profound depth.
Definition
A conjecture is a hypothesis in a formal theory, supported by heuristic evidence, special-case verification, or probabilistic arguments, but lacking a complete proof. Disproving requires a single counterexample. In number theory, analytic methods and computational verification can confirm conjectures for enormous ranges without constituting a proof.
Example
The Riemann Hypothesis: all non-trivial zeros of the Riemann zeta function have real part $1/2$. Verified for the first $10^{13}$ zeros. A proof would determine the distribution of primes with unprecedented precision.
Key Insight
Some conjectures turn out to be independent of ZFC (like certain combinatorial statements), meaning neither they nor their negations can be proven, a possibility that must always be considered.