From: George Murphy (gmurphy@raex.com)
Date: Sun Aug 03 2003 - 16:32:34 EDT
Glenn Morton wrote:
>
> Hi George, I include a note to Burgy at the bottom.
>
> >-----Original Message-----
> >From: George Murphy [mailto:gmurphy@raex.com]
> >Sent: Sunday, August 03, 2003 1:04 PM
> >To: Glenn Morton
> >Cc: asa@calvin.edu
>
> George originally wrote:
>
> >> > There is a basic difference between the Fibonacci sequence
> >> >& the primes. As you
> >> >note, there is a formula with which one can generate as many
> >> >members of the 1st sequence
> >> >as you wish. But there is no general formula for generating
> >> >primes (unless there's been
> >> >a new discovery I haven't heard of, a possibility since I'm hardly
> >> >a number theorist).
> >> >All proposed prime-generating formulas have been found to break
> >> >down at some point.
>
> After my mention of the sieve of Eratosthenes he writes:
>
> > Don't need to - I learned about it from Gamow's _One, Two,
> >Three ... Infinity_
> >when I was about 14. It isn't a prime-generating formula but a device for
> >systematically checking to see if numbers are prime. By a
> >proposed prime-generating
> >formula I mean something like
> > f(n) = n^2 - n + 41
> >which gives primes for n = 1, 2, ... 40 but for n = 41 gives a
> >perfect square.
>
> Ok, there isn't such a simple formula but why does a simple formula mean
> anything? There is a prescription for how to iteratively catch the primes
> and another but simpler formula for catching the numbers in Fibonacci. I
> don't see what 'simple' has to do with it.
Glenn -
Simplicity isn't the issue. There isn't even a complicated formula or
prescription for generating primes. The formula for the Bernoulli numbers, e.g., is a
good deal more involved than that for the Fibonacci sequence
(Bn = (2n)!Z(2n)/2^(2n-1)*pi^2n, where Z is the zeta function) but it's a formula into
which you (or a computer) can plug n = 1,2, 3 .. and generate as many as you wish.
The sieve doesn't work that way. What you're doing with it is seeing if n is prime by
checking multiples of all the integers up to n-1 & if none of them is n then n is prime.
Shalom,
George
George L. Murphy
gmurphy@raex.com
http://web.raex.com/~gmurphy/
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