For fixed integers k and m, with k ≥m ≥ 2, there are only finitely many runs of m consecutive integers with no prime factor exceeding k. We obatin lower bounds for the last such run. Let g(k, m) be its smallest member. For 2 ≤ m ≤ 5 it is shown that g(k, m) > kc logloglogk holds for all sufficiently large k, where c is a constant depending only on m. We also obtain a number of lower bounds with explict ranges of validity. A typical result of this type g(k, 3) > k3 holds just if k ≥ 41.

By employing a precise form of the Hermite interpolation formula we obtain a best possible bound for the number of zeros of p-adic exponential polynomials. As companion to this quantitative result we give a best possible bound on the coefficients, if the exponential polynomial is small at sufficiently many points.

In this paper we use a result from graph theory on the characterization of the line graphs of the complete bigraphs to show that if n is any integer ≥ 2 then any finite linear space having p = n2 − n or p = n2 − n + 1 points, of which at least n2 − n have degree n + 1, and q ≤ n2 + n − 1 lines is embeddable in an FPP of order n unless n = 4. If n = 4 there is only one possible exception for each of the two values of p, and for p = n2 − n, this exception can be embedded in the FPP of order 5.

Sum-free sets may be used to colour the edges of a complete graph in such a way as to avoid monochromatic triangles. We discuss the automorphism groups of such graphs. Embedding of colourings is considered. Finally we illustrate a way of constructing colourings using block designs.

A fundamental region for the reduction of positive quaternary quadratic forms is exhibited. It is a convex polyhedral cone with twelve edges in the 10-dimensional space of quaternary quadratic forms.

Let p be a prime. This paper deals with solutions of functional equations in either formal Laurent series or in analytic functions. Examples connected to special modular functions are considered.

An orthogonal design of order n and type (s1, s2) on the commuting variables x1, x2 is a matrix of order n with entries from {0, ± x1, ± x2} whose row vectors are formally orthogonal.

This note uses cyclotomy to construct orthogonal designs and finds several infinite families of new designs.