Of Conjectures

... And BIG NUMBERS

Funny comix:

Clay Mathematics Institute: We give up

http://abstrusegoose.com/a/210.htm

Search for counter examples to conjectures

Mersenne conjectures

http://en.wikipedia.org/wiki/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

http://en.wikipedia.org/wiki/Euler's_sum_of_powers_conjecture

Check the counterexamples for k=4 and k=5

Chebyshev Bias: Not very large, but large enough

http://mathworld.wolfram.com/ChebyshevBias.html

http://oeis.org/A096630

Archimedes' cattle problem : A large number

http://en.wikipedia.org/wiki/Archimedes'_cattle_problem

(diophantine equations with huge minimal solutions)

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

http://en.wikipedia.org/wiki/Goldbach's_conjecture#Verified_results

No counter example found.

Borsuk's conjecture

http://en.wikipedia.org/wiki/Borsuk's_conjecture

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


Pólya conjecture

http://en.wikipedia.org/wiki/P%C3%B3lya_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

http://en.wikipedia.org/wiki/Scrooge_McDuck#Wealth

Skewes' number

http://en.wikipedia.org/wiki/Skewes'_number

no actual counter-example has been found.

Mertens conjecture

http://en.wikipedia.org/wiki/Mertens_conjecture

Other examples can be found here

http://mathoverflow.net/questions/15444/the-phenomena-of-eventual-counterexamples