[math-fun] Freeman Dyson responds to the question "What do you think is true even though you cannot prove it"? ... order. Now my statement is: it never happens that the reverse of a power of two is a power of five. ...
I find Dyson's response interesting. He claims that he has a statement that is true and unprovable. However, he only explains why he *believes* that the statement is true. This is different from Godel, who has statements that he *knows* to be true. Dyson's arguments that the statement is 1) true, and 2) unprovable, are heuristic. Perhaps the arguments can be made rigorous? He says that "The digits in a big power of two seem to occur in a random way without any regular pattern." This is the basis for the whole thing. Is there a proof of this? I don't see why this is necessarily true, although I can't think of a reason why not. I would naively have thought that the first digit in the primes was uniformly distributed, because I can't think of a reason why there are more primes beginning with a 1 than with a 9, but this is not true. (Anyone got a heuristic argument for this?) Gary McGuire