**Kirill Shikhaev** was born in 1925. Professor, Doctor of Technical Sciences, Academician of the International Informatization Academy. Has held the positions of Director of the Control Systems Research Centre at the Institute of Economics and Complex Communications Issues. For a number of years directed the expert board of the Higher Attestation Committee for control systems, computer technology and informatics.

Was a Chief Engineer for the group of ministries of the military–industrial complex for control systems, and Chief Engineer of the Almaz special system of the USSR State Planning Committee.

Kirill Shikhaev is the author of more than 100 scientific treatises and inventions, the most important of witch are devoted to the theory of differences. His main monographs are The New Arithmetic’s of Second Order Differences, Introduction to the Theory of Differences of Parallel Computing Processes, The Differences Algorithms of Parallel Computing Processes, The Differential Model of the Numbers and others. Has been awarded the Order of October Revolution and of the Patriot War. Winner of the State Prize of the USSR, the Prize of the USSR Council of Ministers and the International Prize named after academian Ivan Yuvishin.

**Professor Victor A. Anokhin**.

Doctor of Technical Sciences, Academician of the International Academy of Engineering. Founder of a scientific field called «Ensuring climatic Safety and Comfort in Subway». Russia is shifting assuredly to the new forms of education. The new specialized educational organizations such as lyceums with the mathematical bias are establishing.

**ARITHMETIC OF DOUBLE DEGREE WITH NUMERICAL SERIES IN THE SOLUTION OF ALGEBRAIC EQUATIONS**

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In the article we propose a new method of solving algebraic equations, based on the arithmetic of double degree with numerical series, which provides the solution of algebraic equations for n>4 by radicals.

**1. A**** few words about traditional "solvers" **of algebraic equations that are studied in every school. The traditional solvers of algebraic equations had called a lot of questions since they were invented in the XV century.

Thus, in the book of L.A. Kaluzhnin, V.I. Sushansky "Transformations and permutations" (Moscow: Nauka, 1979) on page 93 the autho rs put questions to the Cardano formula:

The square roots in the Cardano formula lose their meaning and three mentioned roots are not expressed by this formula.

• Over past six centuries, despite a number of fundamental researches, the problem has not been fully resolved. For example, J.L.Lagrange (1736-1813) questioned, but did not rule out the possibility of solving equations of degree higher than four by radicals. In 1826 the Norwegian mathematician N.G. Abel proved that the algebraic equations of degree higher than four cannot have a solution by radicals. A further research in this area was conducted by E. Galois (1811-1832). In his works Galois did not exclude that some of the equations of high degree with numerical coefficients may be solvable by radicals. Our arithmetic of double degree with numerical series copes with algebraic equations that are still considered to be insoluble by radicals. Our method does not have sticking out "ears of the quadratic trinomial", and all the equations are solved in the given degree.

• It's time to consider this subject once again and to provide the traditional solvers of algebraic equations at least (so far) with new "assistants", which are described in subsequent paragraphs in details.

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