THE 95TH ANNIVERSARY OF VASSILIY ILYICH NECHAEV

The two major goals of the authors are to describe the principal stages of the life of Professor of the Moscow State Pedagogical University Vassiliy Ilyich Nechaev and to give a brief analysis of his scientific and educational activities that has been very influential in the development of both number theory and methods of teaching mathematics at colleges for teacher education. Particular attention is given to the research of Professor V. I. Nechaev and his disciples in analytic number theory and its applications. V. I. Nechaev, being a student of Professor M. K. Grebencha, was a leading figure at the scientific school and the organizer of the analytic number theory research seminar at the Moscow State Pedagogical University. Many of his students received their PhD degrees and later one of them awarded a habilitation in pedagogical sciences. Vassiliy Ilyich Nechaev has published a large number of scientific and methodical works, was a translator of several fundamental scientific monographs. He served as a member of the editorial board of the journal "Mathematical Notes" and as a member of the program committees of several international conferences on algebra and number theory held at the Lev Tolstoy Tula State Pedagogical University.

number theory, cryptography, methods of teaching

1. Nechaev, V. I. 1949, "The representation of integers by sums of terms of the form x(x+1)...(x+n−1) n! " , Doklady Akad. Nauk SSSR (N.S.), vol. 64, pp. 159–162. (Russian)

2. Nechaev, V. I. 1951, "Waring’s problem for polynomials" , Trudy Mat. Inst. Steklov, Moscow, vol. 38, № 1. pp. 190—243. (Russian)

3. Nechaev, V. I. 1953, "On the representation of natural numbers as a sum of terms of the form x(x+1)...(x+n−1) n! " , Izvestiya Akad. Nauk SSSR. Ser. Mat., vol. 17, № 6, pp. 485–498. (Russian)

4. Nechaev, V. I. 1963, "The group of non-singular matrices over a finite field, and recurrent sequences" , Dokl. Akad. Nauk SSSR, vol. 152, pp. 275–277. (Russian)

5. Nechaev, V. I. 1964, "A best-possible estimate of trigonometric sums for recurrent functions with non-constant coefficients" , Dokl. Akad. Nauk SSSR, vol. 154, pp. 520–522. (Russian)

6. Nechaev, V. I. & Polosuev, A. M. 1964, "The distribution of non-residues and primitive roots in a sequence which satisfies a finite-difference equation with polynomial coefficients" , Vestnik Moskov. Univ. Ser. I Mat. Meh., no. 6, pp. 75-–84. (Russian)

7. Nechaev, V. I. & Stepanova, L. L. 1965, "The distirbution of nonresidues and primitive roots in recurrence sequences over a field of algebraic numbers" , Uspehi Mat. Nauk, vol. 20, no. 3 (123), pp. 197-–203. (Russian)

8. Nechaev, V. I. 1968, "Linear recurrent congruences with periodic coefficients" , Mat. Zametki, vol. 3, № 6, pp. 625—632. (Russian)

9. Nechaev, V. I. 1971, "Recurrent sequences" , Algebra and number theory. Moscow. Gos. Ped. Inst. Ucen. Zap., no. 375, pp. 103—123. (Russian)

10. Nechaev, V. I. 1971, "Linear congruences with respect to a power of a prime ideal, and linear recurrent sequences" , Algebra and number theory. Moskov. Gos. Ped. Inst. Uchen. Zap., no. 375, pp. 124—132. (Russian)

11. Nechaev, V. I., 1972, "Trigonometric sums for recurrent sequences" , Dokl. Akad. Nauk SSSR, vol, 206, pp. 811–814. (Russian)

12. Nechaev, V. I. 1972, "Trigonometric sums for recurrent sequences of elements of a finite field" , Mat. Zametki, vol. 11, pp. 597-–607. (Russian)

13. Nechaev, V. I. 1975, "An estimate of the complete rational trigonometric sum" , Mat. Zametki, vol. 17 (1975), no. 6, pp. 839—849. (Russian)

14. Nechaev, V. I. 1975, "Numeric system [textbook for students of physics and mathematics departments of pedagogical institutes]" , Prosveshenie, Moscow, 198 p. (Russian)

15. Nechaev, V. I. 1976, "On the question of representing natural numbers by a sum of terms of the form x(x+1). . . (x+n?1)/n!. Number theory, mathematical analysis and their applications. Trudy Mat. Inst. Steklov., vol. 142 (1976), 195–197, 270. (Russian)

16. Nechaev, V. I. & Topunov, V. L. 1981, "Estimation of the modulus of complete rational trigonometric sums of degree three and four" , Analytic number theory, mathematical analysis and their applications. Trudy Mat. Inst. Steklov, vol. 158, pp. 125–129, 229. (Russian)

17. Nechaev, V. I. 1983, "Ivan Matveevich Vinogradov" , Mat. v Shkole, no. 3, pp. 79–80. (Russian)

18. Nechaev, V. I. 1988, "On the least upper bound on the modulus of complete trigonometric sums of degrees three and four" , Investigations in number theory, pp. 71–76, Saratov. Gos. Univ., Saratov. (Russian)

19. Kubilis, I. P., Nechaev, V. I., Stechkin, S. B. & Shidlovskii, A. B. 1991, "Aleksandr Adol’fovich Bukhshtab" , Uspekhi Mat. Nauk, vol. 46, no. 1(277), pp. 201–202. (Russian); translation in Russian Math. Surveys 46 (1991), no. 1, 241–243

20. Nechaev, V. I. 1992, "On solutions of linear congruences" , New trends in probability and statistics, Vol. 2 (Palanga, 1991), pp. 69—71, VSP, Utrecht

21. Nechaev, V. I. 1994, "On the complexity of a deterministic algorithm for a discrete logarithm" , Mat. Zametki, vol. 55, no. 2, 91–101, 189; translation in Math. Notes 55, no. 1–2, pp. 165–172. (Russian)

22. Nechaev, V. I. 1994, "The complexity of the discrete logarithm" , Ucen. Trudy MPGU, pp. 46–49. (Russian)

23. Nechaev, V. I. 1996, "Distribution on the part of the period of linear recurring sequences over a finite field" , Abstracts of the III International Conference "Modern Problems in Number Theory and its Applications" , Tula, p. 107. (Russian)

24. Nechaev, V. I. 1996, "Distribution of signs in a sequence of rectangular matrices over a finite field" , Tr. Mat. Inst. Steklova, vol. 218, Anal. Teor. Chisel i Prilozh., pp. 335–342. (Russian); translation in Proc. Steklov Inst. Math. 1997, no. 3 (218), pp. 332—339.

25. Nechaev, V. I. 1999, "Ilementy kriptografii" [Elements of cryptography] Osnovy teorii zashchity informatsii. [Fundamentals of information security] Vysshaya Matematika. [Higher Mathematics] Vyssh. Shkola, Moscow, 110 pp. ISBN: 5-06-003644-8 (Russian)