In this paper we consider Diophantine equations of the form $f(x)=g(y)$ where $f$ has simple rational roots and $g$ has rational coefficients. We give strict conditions for the cases where the equation has infinitely many solutions in rationals with a bounded denominator. We give examples illustrating that the given conditions are necessary. It turns out that such equations with infinitely many solutions are strongly related to Prouhet-Tarry-Escott tuples. In the special, but important case when $g$ has only simple rational roots as well, we can give a simpler statement. Also we provide an application to equal products with terms belonging to blocks of consecutive integers of bounded length. The latter theorem is related to problems and results of Erdős and Turk, and of Erdős and Graham.

We give finiteness results for the shifted power values and polynomial values of Littlewood polynomials.

In this paper, we study upper bounds for the degrees of polynomials with only rational roots. First, we assume that the coefficients are bounded. In the second theorem, we suppose that the primes 2 and 3 do not divide any coefficient. The third theorem concerns the case that all coefficients are composed of primes from a fixed finite set.

In this paper, we show how the subjects mentioned in the title are related. First we study the structure of partitions of $$A \subseteq \{1, \dots , n\}$$ A ⊆ { 1 , ⋯ , n } in k-sets such that the first $$k-1$$ k - 1 symmetric polynomials of the elements of the k-sets coincide. Then we apply this result to derive a decomposability result for the polynomial $$f_A(x) := \prod _{x \in A} (x-a)$$ f A ( x ) : = ∏ x ∈ A ( x - a ) . Finally we prove two theorems on the structure of the solutions (x, y) of the Diophantine equation $$f_A(x)=P(y)$$ f A ( x ) = P ( y ) where $$P(y)\in \mathbb {Q}[y]$$ P ( y ) ∈ Q [ y ] and on shifted power values of $$f_A(x)$$ f A ( x ) .

We prove Skolem's conjecture for the exponential Diophantine equation an+tbn=±cn under some assumptions on the integers a,b,c,t. In particular, our results together with Wiles' theorem imply that for fixed coprime integers a,b,c Fermat's equation an+bn=cn has no integer solution n≥3 modulo m for some modulus m depending only on a,b,c. We also provide a generalization where in the equation bn is replaced by a product b1k1⋯bℓkℓ.

In this paper we address the problem of finding well approximating lattices for a given finite set $A$ of points in ${\mathbb R}^n$. More precisely, we search for $\v{o},\v{d_1}, \dots,\v{d_n}\in \mathbb{R}^n$ such that $\v{a}-\v{o}$ is close to $\Lambda=\v{d_1}\mathbb{Z}+\dots+\v{d_n}\mathbb{Z}$ for every $\v{a}\in A$. First we deal with the one-dimensional case, where we show that in a sense the results are almost the best possible. These results easily extend to the multi-dimensional case where the directions of the axes are given, too. Thereafter we treat the general multi-dimensional case. Our method relies on the LLL algorithm. Finally we apply the least squares algorithm to optimize the results. We give several examples to illustrate our approach.

Extending earlier research of Erdős and Graham, we consider the problem of products of factorials yielding perfect powers. On the one hand, we describe how the representability of $\ell$th powers behaves when the number of factorials is smaller than, equa

In a recent work, we have proposed a novel way to approximate point sets with grids using the LLL algorithm, which operates in polynomial time. Now, we show how this approach can be applied to pattern recognition purposes with interpreting the rate of approximation as a new feature for regularity measurement. Our practical problem is the characterization of pigment networks in skin lesions. For this task we also introduce a novel image processing method for the extraction of the pigment network. Then, we show how our grid approximation framework can be applied with specializing it for the recognition of hexagonal patterns. The classification performance of our approach for the pigment network characterization problem is measured on a database annotated by a clinical expert. Throughout the paper we address several practical issues that may help to apply our general framework to other practical tasks, as well.

In Part I of the present paper the following problem was investigated. Let G be a finite simple graph, and S be a finite set of primes. We say that G is representable with S if it is possible to attach rational numbers to the vertices of G such that the vertices v 1, v 2 are connected by an edge if and only if the difference of the attached values is an S-unit. In Part I we gave several results concerning the representability of graphs in the above sense. In the present paper we extend the results from Part I to the algebraic number field case and make some of them effective. Besides we prove some new theorems: we prove that G is infinitely representable with S if and only if it has a degenerate representation with S, and we also deal with the representability with S of the union of two graphs of which at least one is finitely representable with S.

Given a finite nonempty set of primes S, we build a graph $\mathcal{G}$ with vertex set $\mathbb{Q}$ by connecting x and y if the prime divisors of both the numerator and denominator of x-y are from S. In this paper we resolve two conjectures posed by Ruzsa concerning the possible sizes of induced nondegenerate cycles of $\mathcal{G}$, and also a problem of Ruzsa concerning the existence of subgraphs of $\mathcal{G}$ which are not induced subgraphs.

Discrete tomography deals with tomographic reconstruction of greyscale images for which the set of possible grey levels is discrete and small. Here, we develop a discrete approximate reconstruction algorithm. Our algorithm computes an image that has only grey values belonging to a given finite set. It also guarantees that the difference between the given projections and the projections of the reconstructed discrete image is bounded. The bound, which is computable, is independent of the image size. We present reconstruction experiments for a range of phantom images and a varying number of grey values.

Binary tomography deals with the problem of reconstructing a binary image from its projections. In particular, there is a focus on highly underdetermined reconstruction problems for which many solutions may exist. In such cases, it is important to have a quality measure for the reconstruction with respect to the unknown original image. In this article, we derive a series of upper bounds that can be used to guarantee the quality of a reconstructed binary image. The bounds limit the number of pixels that can be incorrect in the reconstructed image with respect to the original image. We provide several versions of these bounds, ranging from bounds on the difference between any two binary solutions of a tomography problem to bounds on the difference between approximate solutions and the original object. The bounds are evaluated experimentally for a range of test images, based on simulated projection data.

We say that k is a P-integer if the first phi(k) primes coprime to k form a reduced residue system modulo k. In 1980 Pomerance proved the finiteness of the set of P-integers and conjectured that 30 is the largest P-integer. We prove the conjecture assuming the Riemann Hypothesis. We further prove that there is no P-integer between 30 and 10^11 and none above 10^3500.

Chamfer distances play an important role in the theory of distance transforms. Though the determination of the exact Euclidean distance transform is also a well investigated area, the classical chamfering method based upon "small" neighborhoods still outperforms it e.g. in terms of computation time. In this paper we determine the best possible maximum relative error of chamfer distances under various boundary conditions. In each case some best approximating sequences are explicitly given. Further, because of possible practical interest, we give all best approximating sequences in case of small (i.e. 5 by 5 and 7 by 7) neighborhoods.

Tomography is concerned with the reconstruction of images from their projections. In this paper, we consider the reconstruction problem for a class of tomography problems, where the images are restricted to binary grey levels. For any given set of projections, we derive an upper bound on the difference between any two binary images having these projections, and a bound on the difference between a particular binary image and any binary image having the given projections. Both bounds are evaluated experimentally for different geometrical settings, based on simulated projection data for a range of images.

In this paper we derive some irrationality and linear independence results for series of the form ∑n=0∞nvnn! where (vn)n=0∞ is either a non-negative integer sequence with υn = o(log n/log log n) or a non-decreasing integer sequence with vn=o(2n/3).

(1.2) f(x) = an xn (n+ a)! + an−1 xn−1 (n− 1 + a)! + · · ·+ a1 x (1 + a)! + a0 1 a! can have a factor of given degree over the rationals. In 1929 Schur [26], [27] proved that a polynomial of the form (1.2) satisfying (1.1) is irreducible if a = 0 and also if a = 1 unless n + 1 is a power of 2 when it may have a linear factor or n = 8 when it may even have a quadratic factor. Also for a = 2 and many other values of a the polynomial f may have a linear factor. On the other hand, a factor of degree > n/2 of f has a cofactor of degree at most n/2. Therefore we consider the question whether f has a factor of degree k with 2 ≤ k ≤ n/2, which we always assume unless specified otherwise. One of our results reads as follows. Theorem 1.1. Let a and k be integers such that 2 ≤ k ≤ n/2, 0 ≤ a ≤ 3k/2. Let f(x) be given by (1.2) where a0, a1, . . . , an are integers satisfying (1.1). Assume that f(x) has a factor of degree k. Then (n, k, a) ∈ {(6, 2, 3), (7, 2, 2), (7, 2, 3), (7, 3, 3), (8, 2, 1), (1.3) (8, 3, 2), (12, 3, 4), (13, 2, 3), (22, 2, 3), (46, 3, 4), (78, 2, 3)}. We shall show that all the cases listed in (1.3) can indeed be realized by some factorizations. The factorizations for the last two cases are due to

Let \({f(x)=(x-a_1)\cdots (x-a_m)}\), where a 1, . . . , a m are distinct rational integers. In 1908 Schur raised the question whether f(x) ± 1 is irreducible over the rationals. One year later he asked whether \({(f(x))^{2^k}+1}\) is irreducible for every k ≥ 1. In 1919 Pólya proved that if \({P(x)\in\mathbb{Z}[x]}\) is of degree m and there are m rational integer values a for which 0 < |P(a)| < 2−N N! where \({N=\lceil m/2\rceil}\), then P(x) is irreducible. A great number of authors have published results of Schur-type or Pólya-type afterwards. Our paper contains various extensions, generalizations and improvements of results from the literature. To indicate some of them, in Theorem 3.1 a Pólya-type result is established when the ground ring is the ring of integers of an arbitrary imaginary quadratic number field. In Theorem 4.1 we describe the form of the factors of polynomials of the shape h(x) f(x) + c, where h(x) is a polynomial and c is a constant such that |c| is small with respect to the degree of h(x) f(x). We obtain irreducibility results for polynomials of the form g(f(x)) where g(x) is a monic irreducible polynomial of degree ≤ 3 or of CM-type. Besides elementary arguments we apply methods and results from algebraic number theory, interpolation theory and diophantine approximation.

Given positive integers n, and p1,…,pr, we establish a fast word combinatorial algorithm for constructing a word w=w1⋯wn of length n, with periods p1,…,pr, and on the maximal number of distinct letters. Moreover, we show that the constructed word, which is unique up to word isomorphism, is a pseudo-palindrome — i.e. it is a fixed point of an involutory antimorphism.