"The Principle of Shrinkage"

[franktaw at netscape.net]

This doesn't completely satisfy the conditions, because it was never 
believed to be true; but I did just encounter it in an unrelated 
context.

Let a(n) be the nth number which is the product of exactly 3 (not 
necessarily distinct) primes (i.e., the nth 3-almost-prime), and let 
f(n) = a(n)/n.  If you graph f(n) for n up to 10000 (probably quite a 
bit higher), it appears to be converging to something near 3.9.  In 
fact, the limit is infinite.

Franklin T. Adams-Watters


-----Original Message-----
From: eclark at math.usf.edu

Perhaps some of the readers of this list may have a suggestion
concerning a recent question on sci.math.research. The question
arose in a discussion of G. Spenser-Brown's purported recent
proof of the Riemann Hypothesis. In the proof GSB used
a new axiom ("the principle of shrinkage") which Tim Chow
formulated as follows:

  If f:R -> R or f:N -> R is a "naturally occurring function" and
  lim(x->oo) f(x) appears to exist, then it does.

Then Tim asks the question:

Can anyone give an example of a conjecture of the form "lim(x->oo) f(x)
exists" for which there was extensive numerical evidence and which
remained open for some time, but was eventually shown to be false?

For the entire thread that this comes from see
http://groups.google.com/group/sci.math.research/browse_frm/thread/1eee03
282e96b657









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