Andrew Granville's Home Page

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 \( G\).


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 \( (1+x+x^2)^n\).


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 \( t\asymp \log x\).


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 \( p\).


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.


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 \( a\)).