Sunday, March 6, 2011

Of Conjectures


Funny comix:
Clay Mathematics Institute: We give up

Search for counter examples to conjectures

Mersenne 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
(diophantine equations with huge minimal solutions)

Goldbach's Conjecture verified results: verified up to 10^18

No counter example found.

Borsuk's conjecture

The conjecture is not known to fail until n = 298.

Pólya conjecture

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

Skewes' number
no actual counter-example has been found.

Mertens conjecture

Other examples can be found here

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