Final answer:
A conjecture in mathematics is an informed guess based on observations, much like a hypothesis in science, and serves as a starting point for further mathematical investigation and experimentation. It is not yet proven and stands as a conjecture until a formal proof is found, at which point it becomes a theorem.
Step-by-step explanation:
In mathematics, a conjecture is an informed prediction or educated guess that is formed based on existing knowledge and observations. It is a preliminary insight that proposes a new hypothesis which hasn't yet been rigorously tested or proven. Unlike a theorem, which has been proved through a formal mathematical process, a conjecture is more like a starting point for further investigation and experimentation.
A conjecture becomes a theorem when it has been proved to be true. However, until it undergoes a proof, it remains a conjecture. Some conjectures, like the famous Fermat's Last Theorem, stand for years or even centuries before being proved or disproved.
Analogous to a scientific hypothesis, a conjecture acts as a starting point for mathematical exploration. It is subject to scrutiny, debate, and testing within the mathematical community. Through this process, conjectures either evolve into valuable components of mathematical theory or are discarded as false leads.