The definition of conjecture says they are basically 'guesswork': http://www.thefreedictionary.com/conjecture http://www.thefreedictionary.com/conjecture. Some

The definition of conjecture says they are basically 'guesswork': http://www.thefreedictionary.com/conjecture http://www.thefreedictionary.com/conjecture. Someone will say 'I think that there are no values of x,y and z can solve x^n+y^n=z^n for n other than 2' but doesn't prove it. This can be due to lack of time, space, or a supreme insight on the part of the conjecturer[1]. The conjecturer has said some important and correct things in the past, so many people will take this experience and look at the conjecture and say 'this is probably right because this guy was right before and we see no errors with this conjecture so we will use it in as a theorem', which is to say they will use it without proof because it makes sense. Then, sometimes 100's of years later, somebody does prove (or disproves) the theorem and get's some press, and for some very important conjectures, prizes. If the conjecture is disproven, then all the math based on it falls down. Conjectures interesting because they are puzzles left by people that they can't answer but they want them to be true, so it is up to future generations with their advanced minds, tools and insights to prove them, leading to progress in the field with the advancements of techniques use to prove these conjectures and make them actual theorems. [1] http://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem http://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem