We determine the convolution sums [Formula: see text] and [Formula: see text] for all positive integers [Formula: see text]. We then use these evaluations together with known evaluations of other convolution sums to determine the numbers of representations of [Formula: see text] by the octonary quadratic forms [Formula: see text] and [Formula: see text]. A modular form approach is used.

We determine formulae for the numbers of representations of a positive integer by certain octonary quadratic forms whose coefficients are 1, 2, 3 and 6.

In this paper we show that the subgraph F 3 is disconnected and that for all integers m, we find all integers a and b such that (9m 2 -4)a 2 + 4 and 5b 2 ± 4 are square. It turns out that the set of numbers b comprises the Fibonacci numbers.

In this paper, we examine some properties of suborbital graphs for the Picard group. We obtain edge and circuit conditions, then propose a conjecture for the graph to be forest. This paper is an extension of some results in (Jones et al. in The Modular Group and Generalized Farey Graphs, pp. 316-338, 1991).

In this paper, we give necessary and sufficient conditions for the graph H u,n to be connected and a forest.