### 1989 Publications

#### On complementary decompositions of the complete graph (with Alexandros Moisiadis and Rolf Rees) Graphs and Combinatorics, 5 (1989), 57-61.

Suppose that $$H$$ is a graph on $$V$$ vertices, using half of the possible edges (like a a path of length 3, or a star on 4 vertices). We let $$H^c$$ be the complement of $$H$$ in $$V$$, even when $$V$$ itself is embedded in a larger graph $$G$$. In this paper we study decomposition of complete graphs into copies of $$H$$, such that the complements of those $$H$$ also form a decomposition of $$G$$.

Article

#### On a class of determinants Fibonacci Quarterly, 27 (1989), 153-256.

We resolve a question of Lehmer to find the determinant of matrix involving the coefficients of $$(1+x+x^2)^n$$.

Article

#### On positive integers $$\leq x$$ with prime factors $$\leq t \log x$$ Number Theory and Applications (ed R.A. Mollin) (Kluwer, NATO ASI), 1989, 403-422.

The number of $$y$$-smooth integers up to $$x$$, looks very different depending on whether $$y\ll \log x$$ or $$y\gg \log x$$. Here we study smooth numbers when $$t\asymp \log x$$.

Article

#### Checking the Goldbach Conjecture on a vector computer (with J. Van de Lune and Herman te Riele) Number Theory and Applications (ed R.A. Mollin) (Kluwer, NATO ASI), 1989, 423-434.

We verify the Goldbach conjecture up to $$10^9$$, observing along the way that the frequency with which $$p$$ is the minimal prime for which $$2n-p$$ is also prime, is not a decreasing function of $$p$$.

Article

#### Limitations to the equi-distribution of primes I (with John B. Friedlander) Annals of Mathematics, 129 (1989), 363-382.

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 here.

Article

#### Least primes in arithmetic progressions Théorie des nombres / Number Theory (ed. J.-M. De Koninck and C. Lévesque) (de Gruyter: New York), 1989, 306-321.

We show, assuming the prime $$k$$-tuplets conjecture, that the primes in arithmetic progressions $$a\pmod q$$, up to $$b\phi(q)\log q$$, are Poisson distributed (for fixed $$a,b$$ as $$q$$ varies over integers coprime to $$a$$).

Article