From: Kurt Foster
Subject: Re: conjectures that are true for large n but are actually false
Date: 24 Jan 1999 19:52:14 GMT
Newsgroups: sci.math
Keywords: More examples of the "Law of Small Numbers"
In , Nelson G. Rich said:
. I recall sometime ago reading about some conjecture that was true for
. all n <= some very large integer L, but false for L. It was quite
. elementary to understand...may have been number theory.
Let's see. There's "2^(2^n) + 1 is prime for every n >= 0". It's true
for n = 0, 1, 2, 3, 4. But it's false for n = 5, 2^32 + 1 being divisible
by 641. Now granted, 5 isn't large, but 2^32 + 1 is respectably large.
There's "n^2 - n + 41 is prime for every positive integer n". That was
never seriously conjectured, but is often given as a cautionary example
against hasty generalization. It holds for n = 1 to 40, but when n = 41
the expresssion is obviously 41^2, which is composite.
There's the conjecture that, if N = 2^p - 1 is prime, then 2^N - 1 is
also prime. The first counterexample is p = 13; 2^13 - 1 = 8191 is
prime, but 2^8191 - 1 isn't.
There's the problem of the changes in sign of Li(x) - pi(x). I don't
know whether the exact location of the first sign change is known even
now, but Littlewood proved that the sign changed infinitely often, and the
first numerical estimate [by Skewes] showed that the sign changed for some
x =< 10^(10^(10^34)).
There is the "Mertens conjecture" that M(x) < sqrt(x) for x > 0, where
M(x) is the sum of the Mobius function mu(n) over n =< x. The first
counterexample was found in the early 1980's, and was a respectably large
number.