> That's certainly the 'obvious' way of trying to construct such starting
> values. Can anyone think of any other ways to come up with them? It
seems
> 'likely' that if there is no finite covering set of divisors there are an
> infinite number of primes... but my intuition is more likely a negative
> endorsement than a positive one ;-)
This is indeed a great unknown, but a familiar situation. Our heuristics and
our instincts tell us we expect an infinite number of Mersenne (and a finite
number of Fermat) primes. Neither of which are proven... perhaps may never
be.
The important thing about constructing such a provable sequence of
composites is it shows that 'expecting an infinity' doesn't necessarily
'guarantee any'. The constructions may seem a little 'artificial', but
they're valid 'islands' of counterexamples in a sea of subjects where
heuristics is currently the best we have.
As for the finiteness of the covering set, again it's currently the limit of
our knowledge. Which is more likely, the existence of an infinite covering
set, or of a single prime?
Chris Nash
Lexington KY
UNITED STATES
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