Sums of unit fractions

Erdos and Straus conjectured that for any positive integer the equation has a solution in positive integers , and . Let k \geq 3$"> and

We show that parametric solutions can be used to find upper bounds on where the number of parameters increases exponentially with . This enables us to prove

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This improves upon earlier work by Viola (1973) and Shen (1986), and is an ``exponential generalization' of the work of Vaughan (1970), who considered the case .

     

Hilbert cubes in progression-free sets and in the set of squares

We present a new method to give upper bounds on the dimension of Hilbert cubes in certain sets. As a special case we improve Hegyvári and Sárközy’s upper bound O((logN)^1/3) for the maximal dimension of a Hilbert cube in the set of squares in [1,N] to O((log logN)²).     

On gaps between quadratic non-residues in the Euclidean and Hamming metrics

The authors have recently introduced and studied a modification of the classical number theoretic question about the largest gap between consecutive quadratic non-residues and primitive roots modulo a prime p$p$, where the distances are measured in the Hamming metric on binary representations of integers. Here we continue to study the distribution of such gaps. In particular we prove the upper bound

_?p≤(0.117198…+o(1))logp/log2${?}_p\le (0.117198\dots +o\left(1\right))logp/log2$

for the smallest Hamming weight _?p${?}_p$ among prime quadratic non-residues modulo a sufficiently large prime p$p$. The Burgess bound on the least quadratic non-residue only gives _?p≤(0.15163…+o(1))logp/log2${?}_p\le (0.15163\dots +o\left(1\right))logp/log2$.     

On Romanov’s constant

Mathematische Zeitschrift - We show that the lower density of integers representable as a sum of a prime and a power of two is at least 0.107. We also prove that the set of integers with exactly...     

Multiplicative decomposability of shifted sets

The following two problems are open.

1. Do two sets of positiveintegers and exist, with at leasttwo elements each, suchthat + coincides with the set of primes for sufficiently largeelements?
2. Let ={6, 12, 18}. Is there an infinite set of positiveintegerssuch that +1? A positive answer would imply that thereare infinitelymany Carmichael numbers with three prime factors.

In this paper we prove the multiplicative analogue of the firstproblem, namely that there are no two sets and , with at leasttwo elements each, such that the product coincides with anyadditively shifted copy +c of the set of primes for sufficientlylarge elements. We also prove that shifted copies of sets ofintegers that are generated by certain subsets of the primescannot be multiplicatively decomposed.     

On variants of the larger sieve

Gallagher's larger sieve is a powerful tool, when dealing with sequences of integers that avoid many residue classes. We present and discuss various variants of Gallagher's larger sieve. This revised version was published online in June 2006 with corrections to the Cover Date.     

Sudoku im Mathematikunterricht

Zusammenfassung Sudoku ist ein immer popul?rer werdendes R?tsel. Wir sind der Meinung, dass es sich hervorragend für den Mathematikunterricht auf verschiedenen Niveaustufen eignet: einerseits zum Trainieren elementarer Logik, andererseits aber (und hier liegt unser Schwerpunkt) zur Abstraktion ausgehend von Konkretem. Schüler k?nnen anhand von Beispielen eigenst?ndig L?sungstrategien entdecken und, unter Anleitung, als allgemeines Prinzip formulieren. Wir stellen zun?chst das R?tsel vor, leiten systematisch L?sungstechniken her und zeigen an Beispielen, dass damit auch recht schwere Sudokus gel?st werden k?nnen. Dann stellen wir Hintergrundinformation zur Verfügung und geben Hinweise zu weiterführenden Informationsquellen. Weiterhin diskutieren wir eine neue Sudokuvariante mit hoher Symmetrie und eine M?glichkeit, für ein gegebenes Gitter die minimal notwendige Anzahl von Hinweisen abzusch?tzen. Mathematics Subject Classification (2000) 05B15 , 97-01     

The distribution of sequences in residue classes

We prove that any set of integers with lies in at least many residue classes modulo most primes . (Here is a positive constant.) This generalizes a result of Erdos and Ram Murty, who proved in connection with Artin's conjecture on primitive roots that the integers below which are multiplicatively generated by the coprime integers (i.e. whose counting function is also ) lie in at least residue classes, modulo most small primes , where as .