The number of fugitive primes
(appendix to "Rational torsion of prime order in elliptic curves over number fields," by Sheldon Kamienny and Barry Mazur )
Astérisque, 228 (1995), 81-100
We show that there are few exceptional primes to the construction in the article of Kamienny and Mazur
We discuss Cramer's contributions to analytic number theory in a modern context, in particular showing that if one believes Cramer's heuristic that there are infinitely many pairs of consecutive primes p, q for which q-p\geq \{ 1+o(1)\} (\log p)^2, then one should believe that there are infinitely many pairs of consecutive primes p, q for which q-p\geq \{ 2e^{-\gamma}+o(1)\} (\log p)^2.
We indicate when the Diophantine equations in the title have finitely many solutions. For example if 1/p+1/q+1/r<1 then Ax^p+By^q =Cz^r has finitely many solutions
It is shown that the complete digraph of type k and order n has a Hering configuration if and only if n\equiv 1 \pmod k, provided n is sufficiently large.
We look at how having three solutions to ax^p+by^p=cz^p for given a,b,c, implies that there are non-diagonal solutions to r^p+s^p+t^p=u^p+v^p+w^p, and we expect that there are only finitely many solutions to this latter equation
We survey the surprising results that the number of primes in intervals and in arithmetic progressions, can be out from the expected number by a constant factor, stemming from Maier's matrix method