Roger B. Eggleton

Colouring the real line

The problem of colouring the real line so that the distance between like coloured numbers does not lie in some specified set D, called the distance set, is discussed. In particular, the minimum number of colours needed for various distance sets are determined.

Colouring prime distance graphs

Four colours are necessary and sufficient to colour all the integers so that any two with difference equal to a prime have different colours. We investigate the corresponding problem when the setD of prescribed differences is a proper subset of the primes. In particular, we prove that ifD contains {2, 3} and also contains a pair of twin primes (one of which may be 3), then four colours are necessary. Numerous results regarding periodic colourings are also obtained. However, the problem of characterizing those setsD which necessitate four colours remains open.

Euclidean Quadratic Fields

algebraic field Q(a) is the smallest subfield of the complex numbers which contains a. The algebraic integers I(a) are those elements of Q(a) which are roots of monic polynomials with (ordinary) integer coefficients. Computation in I(a) is in general unlike computation in the integers Z, since usually there is no analogue of the uniqueness of prime factorization. Historically the (false) assumption that prime factorization is unique in every algebraic field proved to be a stumbling block for various distinguished mathematicians, among them Gabriel Lame. (A nice discussion is given by Edwards [6, Chap. 4].) The problem of determining the algebraic fields which do have unique factorization is still not completely solved. However, in certain fields, known as Euclidean fields, it is possible to define an analogue of Euclids algorithm, and in such cases this guarantees unique factorization. The algebraic fields of degree 2 which have this property are called Euclidean quadratic fields. Work of Davenport and others, culminating in 1952, showed that there are just 21 of them. The well-known book by Hardy and Wright [8] is a standard reference on Euclidean quadratic fields. In 14 of the 21 cases they present proofs that Q(d ) is Euclidean. The reader naturally wonders whether the proofs in the 7 remaining Euclidean cases are difficult. Hardy and Wright also prove that there are no other Euclidean cases with d 0. Indeed, Q(a) = Q(Vd ) in this case. We call this the quadratic field with discriminant d. If the discriminant is negative, the field is complex, and otherwise real. It is convenient to refer to it as Q(Vd ), and to refer to its set of algebraic integers as iG/d ).

Beatty sequences and Langford sequences

Langford sequences and quasi-Langford sequences are defined and used to shed some light on disjoint covering systems and vice versa. We also formulate two conjectures on quasi-Langford sequences, prove their equivalence, and show that they imply a 1973 conjecture on rational disjoint covering sequences.

Unitary Fractions: 10501

part of which dealt with the sum of the squares of the lengths of such chords. Nelson M. Blachman and L. Scribani also considered the case in which the n chords could be extended to have a point of intersection outside the sphere. The first part of the selected solution shows that there is a chord along the ith coordinate axis if di > d R2. To assure existence of extrema, one should also allow the chord to degenerate to a tangent when di = d2 R2. The analysis in the selected solution now shows that the required configuration can exist whenever (n 1)d2 < nR2, and (U) continues to give the upper bound. The analysis of the lower bound shows that the minimum occurs when there is an index j such that d7 = d2 R2 for all i 1 j and dJ = (n 1)R2 (n 2)d2. This leads to a lower bound of 2 nR2-(n-1)d2 for R < d < n/(n 1)R.

Common factors of integers: A graphic view

Abstract The common factor graph of a set of integers has the integers as vertices, two vertices being adjacent just if they have a proper common factor. Such graphs permit visual interpretation of many common factor properties of sets of integers. A characterization of common factor graphs is given. The common factor graph of P , the set of integers ⩾2, is a diameter 2 graph in which every induced subgraph is a common factor graph, and every common factor graph is isomorphic to an induced subgraph of the common factor graph of P . We discuss the problem of finding the length of the smallest initial segment of P which contains a given finite graph as an induced subgraph. Connected common factor graphs of runs of consecutive integers are considered in detail. Pillai and Brauer proved that there exist runs of n consecutive integers not containing any member coprime to all the rest, precisely when n ⩾17. A new uniform construction is given for this result. The paper concludes with relevant numerical results, including constellations of runs with connected common factor graphs occurring around 151 058 and 771 320.

Explaining Simple Combinatorial Answers

(1986). Explaining Simple Combinatorial Answers. The American Mathematical Monthly: Vol. 93, No. 5, pp. 397-399.