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 \( m\)

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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 prime

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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).

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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.

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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.

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

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

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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.

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