On Fri, Oct 15, 1999 at 01:03:57AM -0800, Gordon Bower wrote:
> Here's another argument - suppose the largest unfactored composite was C. How
> long did it take to determine the factorization (or primality) of C-2? (C-1
> would be even.) Then to factor C would only take a marginally longer amount of
> time than it took for C-2. There is no reason you could not complete the
> factorization of C.
Alas, if Gordon's argument is valid, then every positive integer would be
factored. This is the principle of mathematical induction.
A computer may be in a loop. It has factored C-2 and C-1, and is now
working on C. The task of factoring C may indeed take only marginally longer
than it took for C-2, but the extra time is nonetheless positive.
The next number may be factored as you read this paragraph,
so a journal article saying "Every number below this C has been factored,
but C itself has not been" would become outdated almost
immediately, even if true when submitted.
We can use the same argument on how much water our bodies will hold,
or how much pollution to allow. At any given time, adding a single
molecule may seem safe. But the capacities are finite.
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