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What type of famous equation except diophantine equation such that no algorithm can exist to determine whether there is a solution?

I know that if these equation have a solution, then it could be solving with finite amount of time.

By the way, how long does mathematician tend to give up for solving such equations?

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up vote 2 down vote accepted

The impossibility of solving diophantine equations in general leads to a number of other impossibility results. For example, Richardson's theorem http://en.wikipedia.org/wiki/Richardson%27s_theorem says that in a certain class of functions there is no algorithm to test whether the function is identically $0$.

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