David Hobby wrote: > >> Uh? Really? The last time I read about it, the only > > KNOWN > >> perfect numbers were the few that came from... >> 2^(n-1) (2^n - 1) > Right. I should have written the known part. It was yet unknown if there were odd perfect numbers.
OTOH, IIRC the number of odd perfect numbers is
potentially?
infinite, as there are infinite prime Mersenne numbers.
>> What is the smallest known odd perfect number? > > Why it is: > > 235465427730240065113511519531(snip) > No, it isn't. Do you have any idea about it? Something like "it's between 10^10^100 and 10^10^... (100 times) ... 10^100"
I was not aware they had established an _upper_ bound on it, or indeed that they had proved that there were any odd perfect numbers, though I admit I might have missed such an announcement, as there's an awful lot of stuff to keep up with. The last result I recall that had been established was a statement to the effect that if there are any odd perfect numbers, the smallest has to be at least bigger than <some pretty huge number which I don't recall off the top of my head at the moment>.
In fact, this page (updated Monday) seems to have the latest word on the topic:
<<http://www.utm.edu/research/primes/mersenne/>>
which agrees with what I said about there being no known odd perfect numbers, and if there are, they are awfully big . . .
Alberto Monteiro
PS: I once thought about a computer contest, something like: given an extension of a computer language [like C] where the integer type is unbound and memory is unbound, write a set of functions that use only x charaters [or tokens] such that one of them returns the biggest number in a finite time.
The first step would be the generalization of the power function to the next level: a *** 1 = a, a *** (n+1) = a ^ (a *** n). The next step would be a function that takes multiplication, power, and this superpower as members of a sequence of functions. The next step would be the next generation of this superfunction, and then the generalization of all these generalizations.
Been reading Rudy Rucker's _Infinity and the Mind_, have you?
-- Ronn! :)
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