FOM: elimination of analytic methods in number theory
Olivier Gerard
jacquesg at pratique.fr
Tue Mar 3 19:14:31 EST 1998
At 01:01 +0200 98.03.04, Stephen G Simpson wrote:
> I once sat down and convinced myself that the analytic proofs of some
> classical theorems of analytic number theory go through in WKL_0, and
> hence the theorems have "elementary" proofs in PRA. I'm pretty
> confident about Dirichlet's theorem on distribution of primes, and the
> prime number theorem. Beyond that, I don't know. I haven't gone
> through the new stuff like Wiles' theorem, and I haven't gotten any
> number theorists sufficiently interested to bother about it, either.
>
Patrick Cegielski (of Institut Blaise Pascal, Paris and Fontainebleau Institut
Universitaire de Technologie) claims that Dirichlet Theorem is a consequence of
PRA. The reference I have is (in french)
"Le theoreme de Dirichlet est finitiste" Patrick CEGIELSKI,
Jussieu IBP-LITP Internal Report 92.90 (May 1992)
and is certainly available electronically (I can lookup if anyone is
interested) but he has certainly published that in a refereed journal since
then. I remember he uses Parsons work on Sigma_1 induction.
Olivier
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