Things get nicer at some huge number?
- From: wellsoberlin <wellsoberlin@xxxxxxxxx>
- Date: Fri, 25 Apr 2008 22:30:09 +0000 (UTC)
This message is also posted with links for more information on my blog
at http://www.gyregimble.blogspot.com/
Mathematicians have long noticed that in many fields, theorems have
exceptions for small integers. Some theorems for compact
differentiable manifolds can be proved for n bigger than 4, but things
go haywire for 1, 2, 3, 4, especially 4. The finite simple groups have
all been classified as being in one of several infinite families, with
a finite list of exceptions, the largest being of order less than
10^54. (Well, that is small relative to most numbers!) The largest
exceptional Lie group is a manifold of dimension 248. The prime
counting function finally decides to be bigger than the logarithmic
integral somewhere around 10^316. The smallest Perrin pseudoprime that
is not a prime is 271,441.
Perhaps math gets better behaved for very large integers. This
suggests a conjecture:
THE BIG NUMBER CONJECTURE CONJECTURE (sic)
If P(n) is a mathematical statement with one free variable n that
ranges over the positive integers, then there is a number B_P
depending only on the form of P with the property that, in order to
prove that P(n) is true for all positive integers, it is sufficient to
prove P(n) for all positive integers less than B_P.
Remarks:
a) This is precisely a conjecture that a meaningful conjecture exists.
b) The BNC is not a proper conjecture until I define "mathematical
statement" precisely. Anyway, it may be true for some forms of
statements and not others.
c) B_P has to depend on P because you could replace P(n) by P(f(n))
where f(n) is some slow growing function, such as the greatest integer
in log log n.
d) But the dependence of B_P on P must be on the FORM of P in some
sense (number of quantifiers or some such thing). Otherwise the
conjecture is trivially true.
Charles Wells
professional website: http://www.cwru.edu/artsci/math/wells/home.html
blog: http://www.gyregimble.blogspot.com/
abstract math website: http://www.abstractmath.org/MM//MMIntro.htm
personal website: http://www.abstractmath.org/Personal/index.html
.
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