### 1991 Publications

#### Subdesigns in Steiner Quadruple Systems (with Alan Hartman) Journal of Combinatorial Theory, (Series A) 56 (1991), 239-270.

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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#### A note on sums of primes Canadian Mathematical Bulletin, 33 (1991), 452-454.

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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#### Oscillation theorems for primes in arithmetic progressions and for sifting functions (with John Friedlander, Adolf Hildebrand and Helmut Maier) Journal of the American Mathematical Society, 4 (1991), 25-86.

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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#### The lattice points of an $$n$$-dimensional tetrahedron Aequationes Mathematicae, 41 (1991), 234-241.

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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#### The prime factors of Wendt's binomial circulant determinant, (with Greg Fee) Mathematics of Computation, 57 (1991), 839-848.

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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#### Limitations to the equi-distribution of primes, IV (with John Friedlander) Proceedings of the Royal Society, (Series A), 435 (1991), 197-204.

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

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#### On a paper of Agur, Fraenkel and Klein, Discrete Mathematics, 94 (1991), 147-151.

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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#### On pairs of coprime integers with no large prime factors, Expositiones Mathematicae, 9 (1991), 335-350.

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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