In this paper, we study the sets of integers which are n-th terms of Lucas sequences. We establish lower- and upper bounds for the size of these sets. These bounds are sharp for n sufficiently large. We also develop bounds on the growth order of the terms of Lucas sequences that are independent of the parameters of the sequence, which is a new feature.

This paper is the continuation of Hajdu and Tijdeman (Ramanujan J 66:Article 74, 2025), where we deal with Lucas sequences. Here we study integers represented by integer sequences which satisfy binary recursive relations. In the case of non-degenerate sequences we give an upper bound on the largest index of a zero term and bounds on the growth order of the absolute values of the terms, both only in terms of the two initial values, which is a novel feature. Some of these bounds are best possible apart from a multiplicative constant.

Projection ghosts are discrete arrays of signed values positioned so that their discrete projections vanish for some chosen set of n projection angles. Minimal ghosts are designed to be compact, with no internal pixels having value zero. Here we control the shape, number of boundary pixels and area that each minimal ghost encloses. Binary minimal ghosts and their boundaries can themselves be inflated by tiling copies of themselves to make ghosts with larger sizes and different shapes, whilst still retaining the same set of n zero projection angles. The intricate perimeters of minimal ghosts are formed by three strings of connected pixels that are defined by the minimal projection angles. We show that large changes to the ghost areas can be made whilst keeping the length of their segmented perimeters fixed. These inflated boundary ghosts may prove useful as secure watermarks to embed into digital image data. Boundary ghosts may also help guide the selection of angles used to reconstruct images where the object domain is confined to oval shaped arcs.

The goal of discrete tomography is to reconstruct an unknown function $f$ via a given set of line sums. In addition to requiring accurate reconstructions, it is favourable to be able to perform the task in a timely manner. This is complicated by the presence of ghosts, which allow many solutions to exist in general. In this paper we consider the case of a function $f : A \to \mathbb{R}$ where $A$ is a finite grid in $\mathbb{Z}^3$. Previous work has shown that in the two-dimensional case it is possible to determine all solutions in parameterized form in linear time (with respect to the number of directions and the grid size) regardless of whether the solution is unique. In this work, we show that a similar linear method exists in three dimensions under the condition of nonproportionality. We show that the condition of nonproportionality is fulfilled in the case of three-dimensional boundary ghosts.

Discrete tomography focuses on the reconstruction of functions from their line sums in a finite number d of directions. In this paper we consider functions f : A → R where A is a finite subset of ℤ2 and R an integral domain. Several reconstruction methods have been introduced in the literature. Recently Ceko, Pagani and Tijdeman developed a fast method to reconstruct a function with the same line sums as f. Up to here we assumed that the line sums are exact. Some authors have developed methods to recover the function f under suitable conditions by using the redundancy of data. In this paper we investigate the case where a small number of line sums are incorrect as may happen when discrete tomography is applied for data storage or transmission. We show how less than d/2 errors can be corrected and that this bound is the best possible. Moreover, we prove that if it is known that the line sums in k given directions are correct, then the line sums in every other direction can be corrected provided that the number of wrong line sums in that direction is less than k/2.

In discrete tomography, ghosts represent indeterminate locations of a reconstruction when there is insufficient projection information to admit a unique solution. Our previous work presented maximal ghosts, which are tilings of 2^N connected points of ± 1 values with zero line sums over N directions. These directions are given by the recursion v_n+1=v_n + 2ϵ _n+1v_n-1 with ϵ _n+1∈{-1,1} . By including one additional direction, interior points are cancelled leaving only a thin boundary of ghost errors. Here, we show that a simple modification to this recursion is not possible to generate boundary ghosts in three dimensions. Rather, we present a combination of three different recurrences to achieve this goal. We derive results pertaining to the connectivity, size and structure of these shapes.

We present an algorithm that for any given rectangular grid A∈Z2 and set of directions D computes in linear time the values of any function f:A→R outside the convex hull of the union of the switching domains from its line sums in the directions of D. Moreover, the algorithm reconstructs f completely if there are no switching domains. We present a simpler algorithm in case the directions satisfy some monotonicity condition. Finally, for given A we propose how to choose the set D so that only a small number of directions is needed to reconstruct any f from its line sums in the directions of D.

The reconstruction of an unknown function f from its line sums is the aim of discrete tomography. However, two main aspects prevent reconstruction from being an easy task. In general, many solutions are allowed due to the presence of the switching functions. Even when uniqueness conditions are available, results about the NP-hardness of reconstruction algorithms make their implementation inefficient when the values of f are in certain sets. We show that this is not the case when f takes values in a field or a unique factorization domain, such as R or Z. We present a linear time reconstruction algorithm (in the number of directions and in the size of the grid), which outputs the original function values for all points outside of the switching domains. Freely chosen values are assigned to the other points, namely, those with ambiguities. Examples are provided. (C) 2021 Elsevier B.V. All rights reserved.

Discrete tomography reconstructs an image of an object on a grid from its discrete projections along relatively few directions. When the resulting system of linear equations is under-determined, the reconstructed image is not unique. Ghosts are arrays of signed pixels that have zero sum projections along these directions; they define the image pixel locations that have non-unique solutions. In general, the discrete projection directions are chosen to define a ghost that has minimal impact on the reconstructed image. Here we construct binary boundary ghosts, which only affect a thin string of pixels distant from the object centre. This means that a large portion of the object around its centre can be uniquely reconstructed. We construct these boundary ghosts from maximal primitive ghosts, configurations of 2^N connected binary ( ± 1 ) points over N directions. Maximal ghosts obfuscate image reconstruction and find application in secure storage of digital data.

The reconstruction of an unknown function f from its line sums is the aim of discrete tomography. However, two main aspects prevent reconstruction from being an easy task. In general, many solutions are allowed due to the presence of the switching functions. Even when uniqueness conditions are available, results about the NP-hardness of reconstruction algorithms make their implementation inefficient when the values of f are in certain sets. We show that this is not the case when f takes values in a field or a unique factorization domain, such as or . We present a linear time reconstruction algorithm (in the number of directions and in the size of the grid), which outputs the original function values for all points outside of the switching domains. Freely chosen values are assigned to the other points, namely, those with ambiguities. Examples are provided.

This paper aims to show two things. Firstly the importance of Alan Baker's work on linear forms in logarithms for the development of the theory of exponential Diophantine equations. Secondly how this theory is the culmination of a series of greater and smaller discoveries.

Tomography is the theory behind scans, e.g. MRI-scans. Most common is continuous tomography where an object is reconstructed from numerous projections. In some cases this is not applicable, because the object changes too quickly or is damaged by making hundreds of projections (by X-rays). In such cases discrete tomography may apply where only few projections are made. The present paper shows how number theory helps to provide insight in the application and structure of discrete tomography.

For continuous tomography Helgason and Ludwig developed consistency conditions. They were used by others to overcome defects in the measurements. In this paper we introduce a consistency criterion for discrete tomography. We indicate how the consistency criterion can be used to overcome defects in measurements.

This paper provides a survey of results on the greatest prime factor, the number of distinct prime factors, the greatest squarefree factor and the greatest m-th powerfree part of a block of consecutive integers, both without any assumption and under assumption of the abc-conjecture. Finally we prove that the explicit abc-conjecture implies the Erdős-Woods conjecture for each k>2.

In memoriam: N.G. de Bruijn.In this article we present a survey of his papers on combinatorics. The section titles show its variety.1. Common systems of representatives2. De Bruijn cycles3. The De Bruijn-Erdos theorem from incidence geometry4. Bases for integers5. The BEST theorem6. The De Bruijn-Erdos theorem from graph theory7. Factorizations of finite groups8. Rooted trees in the plane9. Permutations of a given shape10. Covering of graphs by dimers11. Counting (Polya's fundamental theorem, Color designs)12. Penrose tilings (C) 2013 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.

In earlier papers we have developed an algebraic theory of discrete tomography. In those papers the structure of the functions $f: A \to \{0,1\}$ and$f: A \to \mathbb{Z}$ having given line sums in certain directions have been analyzed. Here $A$ was a block in $\mathbb{Z}^n$ with sides parallel tothe axes. In the present paper we assume that there is noise in the measurements and (only) that $A$ is an arbitrary or convex finite set in$\mathbb{Z}^n$. We derive generalizations of earlier results. Furthermore we apply a method of Beck and Fiala to obtain results of the followingtype: if the line sums in $k$ directions of a function $h: A \to [0,1]$ are known, then there exists a function $f: A \to \{0,1\}$ such that itsline sums differ by at most $k$ from the corresponding line sums of $h$.

In this note we prove results of the following types. Let be given distinct complex numbers zj satisfying the conditions |zj|=1,zj≠1 for j=1,…,n and for every zj there exists an i such that zi=zj¯. Then infk∑j=1nzjk≤−1. If, moreover, none of the ratios zi/zj with i≠j is a root of unity, then infk∑j=1nzjk≤−1π4logn. The constant −1 in the former result is the best possible. The above results are special cases of upper bounds for infk∑j=1nbjzjk obtained in this paper.

At a conference in Debrecen in October 2010 Nathanson announced some results concerning the arithmetic diameters of certain sets. He proposed some related results on the representation of integers by sums or differences of powers of 2 and 3. In this note we prove some results on this problem and the more general problem about the representation by linear combinations of powers of some fixed integers.

In this note we prove results of the following types. Let be given distinct complex numbers z_j satisfying the conditions |z_j| = 1, z_j ≠ 1 for j=1,..., n and for every z_j there exists an i such that z_i = z̅_̅j̅. Then inf_k∑_j=1^n z_j^k ≤ - 1. If, moreover, none of the numbers z_j is a root of unity, then inf_k∑_j=1^n z_j^k ≤ - 2/π^3log n. The constant -1 in the former result is the best possible. The above results are special cases of upper bounds for inf_k∑_j=1^n b_jz_j^k obtained in this paper.

Sylvester [Syl] proved in 1892 that a product of k consecutive positive integers x, x + 1, . . . , x + k − 1 greater than k is divisible by a prime exceeding k. It is a generalization of Bertrand’s Postulate that there is a prime among k+1, k+2, . . . , 2k. (Take x = k + 1.) The assumption x > k can not be removed since x = 1 should be excluded. In the present paper we discuss generalizations and variations of Sylvester’s theorem.