We show that that for all \(n\) and all \(m\gg n\), under the obvious necessary conditions, there exists Steiner Quadruple Systems of order \( n\) which contain a subdesign of order \( m\)Article

Assuming the prime \( k\)-tuplets conjecture we show that it is possible to construct an infinite sequence of integers such that the average of any two is primeArticle

The Elliott-Halberstam conjecture originally suggested that the Bombieri-Vinogradov Theorem might hold with the moduli getting as large as \( x/(\log x)^A\). We disproved this in I, and here we show that if the Bombieri-Vinogradov Theorem holds the moduli can only get as large as \( x/\exp((\log x)^{5/11}) \) (which was later improved to \( x/\exp((\log x)^{1/2}) \) with Soundararajan).Article

We develop explicit bounds for the number of lattice points with certain features and apply this to study exceptions to the first case of Fermat's Last Theorem.Article

We calculate the prime factors of Wendt's binomial circulant determinant for each even \( m\leq 200\) , and as a consequence resolve the first case of Fermat's Last theorem for prime exponents \( p\) for which \( mp+1\) is also prime and is not a factor of the Wendt's determinant, for some \( m\leq 200\) which is not divisible by 3.Article

We construct polynomials of any given degree with take either significantly more or significantly less prime values than expectedArticle

We count the number of binary strings in which the possible numbers of successive 0's and 1's are restrictedArticle

We estimate the number of pairs of coprime \( y\)-smooth integers, and get slightly improved results in a number of classical analytic number theory problems.Article