[FOM] Applications of the Riemann hypothesis and the ABC conjecture to independence results

[weiermann@math.uu.nl]

Dear Members of FOM,

I would like to announce the following results
which are inspired by Harvey Friedman's FOM
posting "Urbana thoughts" from June 2000.

There exist two true forall exists Assertions A and B
in the language of arithmetic
and rational numbers a,b,c,d>0 (a>b,c>d) such that:

1. PRA does not prove A(a).
2. If the ABC conjecture is true, then
   PRA does not prove A(b).
3. For positive r small enough PRA does prove A(r).

4. PRA does not prove B(c).
5. If the Riemann conjecture is true, then
   PRA does not prove B(d).
6. For positive s small enough PRA does prove B(s).

Thus in principle independence results can be used
for an attack on unproved hypotheses. This might be
of interest for the Cramer conjecture over the
distribution of primes in short intervals.
(For the Cramer conjecture similar results as above hold.)

The proof of assertions 1,2,4,5 is based on distribution
results for square free numbers and primes in short intervals
(Results by Filaseta, Heath Brown, Granville and Cramer.)

Best regards,
Andreas Weiermann