... And BIG NUMBERS
Clay Mathematics Institute: We give up
Search for counter examples to conjectures
Euler observed that the Mersenne number at M_61 is prime, so refuting the conjecture.
M_61 is a very large number.
Euler's sum of powers conjecture
Check the counterexamples for k=4 and k=5
Chebyshev Bias: Not very large, but large enough
Archimedes' cattle problem : A large number
Goldbach's Conjecture verified results: verified up to 10^18
No counter example found.
The conjecture is not known to fail until n = 298.
An explicit counterexample, of n = 906,180,359 was given by R. Sherman Lehman in 1960. The smallest counterexample is n = 906,150,257, found by Minoru Tanaka in 1980
Big Number: The sum of Scrooge's wealth is very controversial
no actual counter-example has been found.
Other examples can be found here