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Andrew Granville's Home Page

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≫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/(logx)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((logx)5/11) (which was later improved to x/exp((logx)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≤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≤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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