On the diophantine equation (x − 1)(y − 1) = (z − 1)

Let k ≥ 3 be an integer. We study the possible existence of pairs of distinct positive integers (a, b) such that any of the three numbers a + 1, b + 1, and ab + 1 is a k-th power. We further investigate several related questions.     

Selective Encapsulation of Disaccharide Xylobiose by an Aromatic Foldamer Helical Capsule

Xylobiose sequestration in a helical aromatic oligoamide capsule was evidenced by circular dichroism, NMR spectroscopy, and crystallography. The preparation of the 5 kDa oligoamide sequence was made possible by the transient use of acid‐labile dimethoxybenzyl tertiary amide substituents that disrupt helical folding and prevent double helix formation. Binding of other disaccharides was not detected. Crystallographic data revealed a complex composed of a d ‐xylobiose α anomer and two water molecules accommodated in the right‐handed helix. The disaccharide was found to adopt an unusual all‐axial compact conformation. A dense network of 18 hydrogen bonds forms between the guest, the cavity wall, and the two water molecules.     

On the diophantine equation

One of the purposes of this note is to correct the proof of a recent result of Y. Guo & M. Le on the equation . Moreover, we prove that the diophantine equation , , , , , gcd, , has only finitely many solutions, all of which satisfying .

     

Influence of relative humidity on electrical properties of alpha-Al2O3 powders: resistivity and electrochemical impedance spectroscopy

The influence of humidity on the electrical properties of alpha-Al2O3 powders has been investigated using adsorption isotherms, DC resistivity, and electrochemical impedance spectroscopy. Samples of two alpha-Al2O3 particle sizes were examined, both individually and mixed together. The results show that the grain-bed resistivity decreases with humidity, whereas the grain capacitance is almost constant. The resistivity difference between the two particle sizes is of several orders of magnitude, while the capacitance values are not very different. These results are interpreted in terms of the layer-by-layer growth of water adsorbed on the grain surfaces. The first, more tightly bound adsorbed layer does not provoke the same effects as those layers adsorbed at higher relative humidity.     

The Structure of 2‐[(1S)‐(+)‐10‐camphorsulfonamino]‐6‐aminopyridine in the Solid State and in Solution; Unprecedented Cocrystallization of Four Stereoisomers

The solid and solution structures of a new optically active aminopyridine compound, 2‐[(1S)‐(+)‐10‐camphorsulfonamino]‐6‐aminopyridine [(S)‐csaap], 1 , are reported. Crystal data: space group P2₁, a = 8.9729 (5), b = 10.9447 (6), c = 36.693 (2) Å, β = 96.435 (1)°, V = 3580.8 (3) ų, Z = 8, R₁ = 0.0673 and wR₂ = 0.1600 with I > 2σ(I). This chiral compound shows an unprecedented cocrystallization of four stereoisomers, which are characterized by X‐ray crystallography and NMR spectroscopy.     

Syntheses and Structures of Zinc(II) Co‐Ordination Polymers Bridged by Flexible Bidentate Nitrogen Ligands

The reaction of Zn(NO₃)₂·6H₂O and bpp (bpp = 1,3‐Bis(4‐pyridyl)pronpane) in CH₃OH afforded the complex [Zn(bpp)(NO₃)₂]_n, 1 . The IR has been recorded and the structure has been determined. Crystal data for 1: Space group P2(1)/n, a = 11.749(1), b = 11.413(1), c = 11.942(1) Å, β = 96.06(6)°. V = 1592.5(3) ų, Z = 4 with final residuals R1 = 0.0484 and wR2 = 0.0984. The complexes show supramolecular structure in the solid state by intermolecular hydrogen bonding interaction.     

Binding sites of amyloid beta-peptide in cell plasma membrane and implications for Alzheimer's disease

The binding of amyloid beta peptides (Abeta) to plasma membranes appears to be a promising point of intervention in the events leading to the development of Alzheimer's disease (AD). This binding has been studied as regards the direct toxicity of Abeta on neurons, and the activation of a local inflammation phase involving microglia. By virtue of its structure, Abeta is able to bind to a variety of biomolecules, including lipids, proteoglycans and proteins. This review focuses on the membrane proteins that can mediate the interaction between Abeta and the plasma membranes in AD. On neurons, these are APP (amyloid precursor protein), the NMDA-R (N-methyl-D-aspartate receptor), integrins, the alpha7nicotinic acetylcholine receptor (alpha7nAChR), the P75 neurotrophin receptor (P75NTR) and the CLAC-P/collagen type XXV (collagen-like Alzheimer amyloid plaque component precursor/collagen XXV). On glial cells, FPRL1 (formyl peptide receptor-like 1), the scavenger receptors A, BI (SR-A, SR-BI) and CD36, a complex involving CD36, alpha(6)beta(1)-integrin and CD47, and heparan sulfate proteoglycans have been reported to bind Abeta. It should be noted that integrins, RAGE (receptor for advanced glycosylation end-products), the Serpin-enzyme complex receptor (SEC-R) and the insulin receptor can bind Abeta and are present on neurons and on glial cells. After a presentation of the structure and the function of each of these proteins, the method used to prove their binding to Abeta is described, and the implication of this binding in AD is discussed. Finally, it is underlined that multireceptor complexes containing integrins may be involved in this interaction.