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There are NO unsolved theorems. Ok I cheated. Unproved assertions are strictly speaking called "conjectures" or "hypotheses". Only when they are proved should they be called theorems. I don't think you'll find anyone who knows them all but here are some easy to state ones: Goldbach's conjecture: Every even number is the sum of two primes. Twin prime conjecture: There are an infinite number of twin primes (pairs of primes that differ by two) Cayleys conjecture: Starting with a positive number, the repeated process "divide by two if even, multiply by three and add one if odd" will always reach one. and here's a lot more: http://www.absoluteastronomy.com/encyclopedia/u/un/unsolved_problems_in_mathematics.htm
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Not theories -- theorems. There are some famous ones that go back many centuries. Until a proof is found they are better referred to as mathematical conjectures rather than unproven theorems. In fact there is a $1 Million prize for proving any so-called Millennium Problems, offered by the Clay Mathematics Institute. Details here: http://www.claymath.org/millennium/
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