If what the original contributor means is that there is no largest prime
number, then he's right. The formal proof of an infinity of primes was known
in ancient times, certainly as far back as Euclid. It follows that, since
there must be a finite number of primes smaller than or equal to any finite
integer, there must be an infinity of larger primes - not just at least one!
In any case we're not claiming that 2^30402457-1 is _the_ largest prime
number, just the largest anyone knows of at the moment. AFAIK!
On Sunday 22 January 2006 19:12, Steinar H. Gunderson wrote:
> On Sun, Jan 22, 2006 at 07:59:15PM +0100, Schneelocke wrote:
> > I *think* what he means is this:
> >
> > Define a_0 = 13901, b_0 = 5019630, a_{n+1} = a_n * b_n, b_{n+1} = a_n
> > + b_n. Then b_n is conjectured to always be prime (above a certain n,
> > I assume, at least).
>
[ big snip, details unimportant ]
>
> Clearly this conjuncture is not worth very much. :-)
Yes. In fact I think there is a formal proof that _every_ simple algorithm
generating a list of numbers _must_ generate at least some composites. Maybe
even for sufficiently large n that _all_ subsequent terms must be composite?
Can't remember where I saw this but it does look intuitively reasonable...
There may be a dependency on the Riemann hypothesis; if so then the sooner
someone proves Riemann the better - though unfortunately mathematics doesn't
impress everyone; mathematics departments still get letters from nutters
claiming to have invented a method of squaring the circle, despite really
solid and ancient proofs that no such procedure is possible.
Regards
Brian Beesley
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