We obtain sharp lower bounds for the number of solutions to diagonal equations in finite fieldsArticle

We show that there are few exceptional primes to the construction in the article of Kamienny and MazurArticle

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 \( q-p\geq \{ 2e^{-\gamma}+o(1)\} (\log p)^2\).Article

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 solutionsArticle

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.Article

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 Article

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 Article

A review of this rather silly bookArticle