Fifteen consecutive integers with exactly four prime factors

We describe a successful search for a sequence of fifteen consecutive integers, each the product of exactly four prime factors. Fifteen is best possible.

     

An Additive Problem with Primes in Arithmetic Progressions

In this paper, we extend a classical result of Hua to arithmetic progressions with large moduli. The result implies the Linnik Theorem on the least prime in an arithmetic progression.     

Algebraic structure of association schemes of prime order

Finite groups of prime order must be cyclic. It is natural to ask what about association schemes of prime order. In this paper, we will give an answer to this question. An association scheme of prime order is commutative, and its valencies of nontrivial relations and multiplicities of nontrivial irreducible characters are constant. Moreover, if we suppose that the minimal splitting field is an abelian extension of the field of rational numbers, then the character table is the same as that of a Schurian scheme.     

A Note on Rings and Modules Graded by Some Special Groups

In this note we prove that the prime madicals of some graded rings and the scoles of some graded modules are also graded. The results on the relations between the prime radical and the graded prime radical are also presented. Moreover, the relations between a graded ring and its initial subring are studied.     

Hidden structure in the randomness of the prime number sequence?

We report a rigorous theory to show the origin of the unexpected periodic behavior seen in the consecutive differences between prime numbers. We also check numerically our findings to ensure that they hold for finite sequences of primes, that would eventually appear in applications. Finally, our theory allows us to link with three different but important topics: the Hardy–Littlewood conjecture, the statistical mechanics of spin systems, and the celebrated Sierpinski fractal.     

Total synthesis of 3,3′:4′,3″-ter-1,2,5-thiadiazole using a synthetic utility of S4N4·SbCl5 complex

3,3′:4′,3″-Ter-1,2,5-thiadiazole, an useful oligoheterocyclic compound, has been accomplished in seven steps from 1-(5-methyl-3-isoxazolyl)ethanone or diethyl acetylenedicarboxylate using a synthetic utility of tetrasulfur tetranitride antimony pentachloride (S₄N₄·SbCl₅) complex to make a 1,2,5-thiadiazole ring.     

Synthesis, characterization and Schlenk equilibrium studies of methylmagnesium compounds with O- and N-donor ligands - the unexpected behavior of [MgMeBr(pmdta)] (pmdta = N,N,N′,N″,N″-pentamethyldiethylenetriamine)

MgMe₂ (1) was found to react with 1,4-diazabicyclo[2.2.2]octane (dabco) in tetrahydrofuran (thf) yielding a binuclear complex [{MgMe₂(thf)}₂(μ-dabco)] (2). Furthermore, from reactions of MgMeBr with diglyme (diethylene glycol dimethyl ether), NEt₃, and tmeda (N,N,N′,N′-tetramethylethylenediamine) in etheral solvents compounds MgMeBr(L), (L = diglyme (5); NEt₃ (6); tmeda (7)) were obtained as highly air- and moisture-sensitive white powders. From a thf solution of 7 crystals of [MgMeBr(thf)(tmeda)] (8) were obtained. Reactions of MgMeBr with pmdta (N,N,N′,N″,N″-pentamethyldiethylenetriamine) in thf resulted in formation of [MgMeBr(pmdta)] (9) in nearly quantitative yield. On the other hand, the same reaction in diethyl ether gave MgMeBr(pmdta) · MgBr₂(pmdta) (10) and [{MgMe₂(pmdta)}₇{MgMeBr(pmdta)}] (11) in 24% and 2% yield, respectively, as well as [MgMe₂(pmdta)] (12) as colorless needle-like crystals in about 26% yield. The synthesized methylmagnesium compounds were characterized by microanalysis and ¹H and ¹³C NMR spectroscopy. The coordination-induced shifts of the ¹H and ¹³C nuclei of the ligands are small; the largest ones were found in the tmeda and pmdta complexes. Single-crystal X-ray diffraction analyses revealed in 2 a tetrahedral environment of the Mg atoms with a bridging dabco ligand and in 8 a trigonal-bipyramidal coordination of the Mg atom. The single-crystal X-ray diffraction analyses of [MgMe₂(pmdta)] (12) and [MgBr₂(pmdta)] (13) showed them to be monomeric with five-coordinate Mg atoms. The square-pyramidal coordination polyhedra are built up of three N and two C atoms in 12 and three N and two Br atoms in 13. The apical positions are occupied by methyl and bromo ligands, respectively. Temperature-dependent ¹H NMR spectroscopic measurements (from 27 to −80 °C) of methylmagnesium bromide complexes MgMeBr(L) (L = thf (4); diglyme (5); NEt₃ (6); tmeda (7)) in thf-d₈ solutions indicated that the deeper the temperature the more the Schlenk equilibria are shifted to the dimethylmagnesium/dibromomagnesium species. Furthermore, at −80 °C the dimethylmagnesium compounds are predominant in the solutions of Grignard compounds 4-6 whereas in the case of the tmeda complex7 the equilibrium constant was roughly estimated to be 0.25. In contrast, [MgMeBr(pmdta)] (9) in thf-d₈ revealed no dismutation into [MgMe₂(pmdta)] (12) and [MgBr₂(pmdta)] (13) even up to −100 °C. In accordance with this unexpected behavior, 1:1 mixtures of 12 and 13 were found to react in thf at room temperature yielding quantitatively the corresponding Grignard compound 9. Moreover, the structures of [MgMeBr(pmdta)] (9c), [MgMe₂(pmdta)] (12c), and [MgBr₂(pmdta)] (13c) were calculated on the DFT level of theory. The calculated structures 12c and 13c are in a good agreement with the experimentally observed structures 12 and 13. The equilibrium constant of the Schlenk equilibrium (2 9c ? 12c + 13c) was calculated to be K_gas = 2.0 × 10⁻³ (298 K) in the gas phase. Considering the solvent effects of both thf and diethyl ether using a polarized continuum model (PCM) the corresponding equilibrium constants were calculated to be K_thf = 1.2 × 10⁻³ and K_ether = 3.2 × 10⁻³ (298 K), respectively.     

Investigation on the thermal degradation of acrylic polymers with fluorinated side-chains

The thermal degradation of poly-2,2′,3,3′,4,4′,5,5′,6,6′,7,7′,7″-tridecafluoroheptylacrylate and poly-2,2′,3,3′,4,4′,5,5′,6,6′,7,7′-dodecafluoroheptylmethacrylate has been studied in isothermal conditions at 450-750 °C using pyrolysis-gas chromatography. The type and composition of the pyrolysis products give useful information about mechanism of thermal degradation. It was shown that the main thermal degradation process for both polymers is random main-chain scission. The major degradation products for fluorinated polyacrylate are monomer, dimer, saturated diester, trimer, and corresponding methacrylate. The fluorinated polymethacrylate gives monomer as the main product of thermal destruction. As a result of side-chain reaction, the thermal degradation of the fluorinated polyacrylate also produces remarkable amounts of alcohol. On the other hand, the respective alcohol is only a minor component among the pyrolysis products of the fluorinated polymethacrylate. For both polymers, the main nontrivial degradation product coming from the alkyl ester decomposition is the corresponding fluorinated cyclohexane. The formation of the fluorinated cyclohexanes may be accounted for a nucleophilic bimolecular substitution pathway.     

N,N′,N″-Tri-Boc-guanidine (TBG): a common starting material for both N-alkyl guanidines and amidinoureas

In this Letter, we describe the unexpected reaction pattern of N,N′N″-tri-Boc-guanidine (TBG) with amines at room temperature and under reflux conditions affording N-substituted guanidines and amidinoureas, potentially important compounds with extensive applications in medicinal chemistry. This investigation shows that TBG is an excellent, readily available common starting material for the synthesis of various N-alkyl guanidines as well as N-alkyl-N′-substituted amidinoureas by simply manipulating the reaction conditions.     

Order-disorder at Fe sites in SrFe2/3B″1/3O3 (B″=Mo, W, Te, U) tetragonal double perovskites

We have prepared SrFe_2/3B″_1/3O₃ (B″=Mo, U, Te, and W) double perovskites in polycrystalline form by ceramic methods. Phases with B″=U, Te and W have been studied by X-ray powder diffraction and the results have been compared with neutron diffraction data available for B″=Mo. At room temperature, the stoichiometric samples crystallize in the tetragonal crystal system (space group I4/m, Z=4). Cell parameters when B″=U, Te and W are a=5.6936(1) Å, c=8.0637(1)Å; a=5.5776(1) Å, c=7.9144(3) Å and a=5.5707(3) Å, c=7.9081(5) Å, respectively.The Mössbauer spectra at room temperature for all compounds show hyperfine parameters belonging to two Fe³⁺ sites located at lattice positions with different degrees of distortion. This is in agreement with diffraction data that indicate that the series of compounds display different degrees of Fe-site disorder, which increases in the following sequence: Mo<U<Te<W.