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Conjecture is a kind of guesswork: you make a judgment based on some inconclusive or incomplete evidence and you call it a conjecture. Or you make a kind of statement, but this is based only on your opinion, or again, guesswork - this is a conjecture once again. You may be proved right or wrong. Sometimes it may take centuries for people to prove you either right or wrong. If they prove you right, your conjecture will become a theorem (but it will be probably called after the person who solved it!)

However, if you have an idea that you can demonstrate is true, or you can assume to be demonstrable, you've got yourself a true THEOREM. In other words, you must provide a proof, or otherwise persuade the world that you have one.

Have a look at some famous conjectures and theorems, as well as at some problems that have been giving mathematicians a reason to get up in the morning for many years (centuries in some cases!).

Some solved and some unsolved problems from the history of mathematics

23 Problems of Hilbert

Euler's Conjecture

Fermat's Conjecture

Fermat's Last Theorem

Four Colour Theorem

Fundamental Theorem of Algebra

Goldbach's Conjecture

Pythagoras' Theorem

Euclid's Proof of the Infinity of Prime Numbers

Fundamental Theorem of Arithmetic

Hilbert

Euler

Fermat

Pythagoras

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