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

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

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

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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

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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.

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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

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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

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A review of this rather silly book

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