On Waring's Problem for Six Cubes and Higher Degrees

It is proved that the equation $$n = x_{\,1}^3 + x_{\,2}^3 + x_{\,3}^3 + x_{\,4}^3 + x_{\,5}^3 + x_{\,6}^3 + u^4 + v^9$$ has nonnegative integral solutions if $n \equiv 1\left( {\bmod 5} \right)$ is even and sufficiently large. Bibliography: 8 titles.     

Quadratic irrationals with short periods of expansion into continued fraction

We study the class number of an indefinite binary quadratic form of discriminant d based on the expansion of d into a continued fraction and single out sequences of d for which h(d) has a lower-bound extimate. Progress is made for the conjecture on the estimate of the quantity of prime discriminants d with fixed length of period of expansion of d. Bibliography: 15 titles.Dedicated to the 90th anniversary of G. M. Goluzin's birthTranslated fromZapiski Nauchnykh Seminarov POMI, Vol. 237, 1997, pp. 31–45.     

Synthesis of 3,3-disubstituted 2-iminoindolines by cyclization of 1-phenyl-2-acylhydrazines

3,3-Disubstituted 2-iminoindolines and the corresponding 10,10-disubstituted tetrahydropyrimido-[1,2-a] indoles, including compounds with spiran structures, were synthesized by reaction of 1-phenyl-2-acylhydrazines and l-phenyl-2-acylpyrazolidines geminally substituted at the-carbon atom of the acyl residue with phosphorus oxychloride. An instance of cleavage of the methyl group was detected.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 12, pp. 1632–1635, December, 1975.     

On the Pellian Equation

Let (d) be the least solution of the Pellian equation x ²-dy ² = 1. It is proved that there exists a sequence of values of d having a positive density and such that (d) > d ^2-, where is an arbitrary positive constant. Bibliogrhaphy: 7 titles.     

Dimerization of Chlorocuprates in Weakly Polar Solvents

The solutions of CuCl₂ and tributylbenzylammonium chloride in chlorobenzene, which are used as catalysts for a number of processes with the participation of halogenated hydrocarbons, were studied by electronic spectroscopy in visible and UV regions. It was found that copper(II) ions occurred as two anionic chlorocuprate species (CuCl^2- ₄ and Cu₂Cl^2- ₆), which were in equilibrium, in solutions over wide ranges of concentrations and temperatures. The individual spectra of the two above chlorocuprate species were distinguished. The thermodynamic parameters of the equilibrium 2CuCl^2- ₄ Cu₂Cl^2- ₆ + 2Cl^– were calculated.     

The Spectrum of Lévy Constants for Quadratic Irrationalities and Class Numbers of Real Quadratic Fields

Let h(d) be the class number of the field and let be the Lévy constant. A connection between these constants is studied. It is proved that if d is large, then the value h(d) increases, roughly speaking, at the rate as grows. A similar result is obtained in the case where the value is close to , i.e., to the least possible value. In addition, it is shown that the interval contains no values of for prime p such that p 3 (mod 4). As a corollary, a new criterion for the equality h(d)=1 is obtained. Bibliography: 14 titles.