FOM: elimination of analytic methods in number theory

Olivier Gerard jacquesg at
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.


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