As mathematicians study they often run across results that they are sure are true, but that they can not (yet) prove. These results are called conjectures. To be called a conjecture (as opposed to an open question), there should be some strong evidence or heuristics supporting the result. See the law of small numbers for the most common trap beginning conjecturers could fall into.
Some conjectures stand for hundreds of years before they are finally proven (or disproven). For example, Fermat's Last Theorem stood as a conjecture for 350 years before Wiles proved it. Perhaps you will be the one to prove one of the conjectures on the page(s) linked below!
Related pages (outside of this work)