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Complexity questions in number theory

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Abstract

An analysis of the complexity of the known solution algorithms is given for four problems of number theory — the solving of Diophantine equations and inequalities and the seeking of Diophantine approximations and solutions of quadratic Diophantine equations. A comparison is made of the various algorithms on the basis of their time complexity. The relation of time complexity to the sizes of the intermediate numbers is particularly stressed. A machine independent description of complexity classes is given and some open problems are formulated.

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Literature cited

  1. É. T. Avanesov and K. K. Billevich, “On the fundamental units of cubic fields of positive discriminant,” Mat. Zametki,29, No. 6, 801–811 (1981).

    Google Scholar 

  2. I. V. Arnol'd, Number Theory [in Russian], Uchpedgiz (1939).

  3. Z. I. Borevich and I. R. Shafarevich, Number Theory [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  4. B. A. Venkov, Elementary Number Theory [in Russian], ONTI, Moscow-Leningrad (1937).

    Google Scholar 

  5. I. M. Vinogradov, Fundamentals of Number Theory [in Russian], Nauka, Moscow (1965).

    Google Scholar 

  6. G. F. Voronoi, On a Generalization of the Continued Fraction Algorithm [in Russian], Warsaw (1896).

  7. G. F. Voronoi, Collected Works [in Russian], Vols. 1–3, Kiev (1952).

  8. B. N. Delone, “Geometry of positive quadratic forms,” Usp. Mat. Nauk,3, 16–62 (1937),4, 102–164 (1938).

    Google Scholar 

  9. G. N. Delone and D. K. Faddeev, “Theory of third-degree irrationalities,” Tr. Mat. Inst. Steklov. Akad. Nauk SSSR,11 (1940).

  10. R. M. Karp, “Reducibilities among combinatorial problems,” in: Complexity of Computer Computations, R. Miller and J. Thatcher (eds.), Plenum Press, New York (1972), pp. 85–104.

    Google Scholar 

  11. J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge Univ. Press (1957).

  12. J. W. S. Cassels, An Introduction to the Geometry of Numbers, Springer-Verlag, Berlin (1959).

    Google Scholar 

  13. M. K. Kerimov, “On methods for computing the Riemann zeta-function and certain generalizations of it,” Zh. Vychisl. Mat. Mat. Fiz.,20, No. 6, 1580–1597 (1980).

    Google Scholar 

  14. S. A. Cook, “The complexity of theorem-proving procedures,” Proc. Third Ann. ACM Sympos. Theory of Computing, Assoc. for Comput. Mach., New York (1971), pp. 151–158.

    Google Scholar 

  15. L. A. Levin, “Universal sorting problems,” Probl. Peredach. Inf.,9, No. 3, 115–116 (1973).

    Google Scholar 

  16. Yu. V. Linnik, Selected Works. Number Theory, Ergodic Methods, and L-Functions [in Russian], Nauka, Leningrad (1979).

    Google Scholar 

  17. Yu. V. Linnik, Selected Works. Number Theory, L-Functions and the Variance Method [in Russian], Nauka, Leningrad (1980).

    Google Scholar 

  18. A. I. Mal'tsev, Algorithms and Recursive Functions [in Russian], Fizmatgiz, Moscow (1965).

    Google Scholar 

  19. A. V. Malyshev, “On the representation of integers by positive quadratic forms,” Tr. Mat. Inst. Steklov. Akad. Nauk SSSR,15 (1962).

  20. K. L. Manders and L. Adleman, “NP-complete problems of solving quadratic equations with two unknowns,” Kibern. Sb., Nov. Ser., No. 17, Mir, Moscow (1980), pp. 124–153.

    Google Scholar 

  21. Yu. I. Manin, Computable and Incomputable [in Russian], Sov. Radio, Moscow (1980).

    Google Scholar 

  22. S. V. Pakhomov, “Machine-independent description of certain machine complexity classes,” J. Sov. Math.,20, No. 4, 2358–2363 (1982).

    Google Scholar 

  23. E. V. Podsypanin, “A generalization of the continued fraction algorithm related to the Viggo Brunn algorithm,” J. Sov. Math.,16, No. 1, 885–893 (1981).

    Google Scholar 

  24. K. Prakhar, Distribution of Primes [Russian translation], Mir, Moscow (1967).

    Google Scholar 

  25. M. A. Frumkin, “Application of module arithmetic to the construction of solution algorithms for linear equation systems,” Dokl. Akad. Nauk SSSR,229, No. 5, 1067–1070 (1976).

    Google Scholar 

  26. M. A. Frumkin, “Complexity of discrete problems,” Preprint, TsÉMI, Moscow (1981).

    Google Scholar 

  27. H. Hasse, Number Theory, Springer-Verlag, Berlin (1980).

    Google Scholar 

  28. H. G. Zimmer, “Computational problems, methods, and results in algebraic number theory,” in: Matematika, Novoe v Zarubezhnoi Nauke, No. 2, Mir, Moscow (1976), pp. 221–298.

    Google Scholar 

  29. L. Adleman and K. Manders, “Reducibility, randomness, and intractability (abstract),” Conf. Record Ninth Ann. ACM Sympos. Theory of Computing, Assoc. Comput. Mach., New York (1977), pp. 151–163.

    Google Scholar 

  30. L. Adleman and K. Manders, “Diophantine complexity,” 17th Ann. Sympos. Found. Comput. Sci., IEEE Computer Soc., Long Beach, CA (1976), pp. 81–88.

    Google Scholar 

  31. L. M. Adleman and K. Manders, “Reductions that lie,” 20th Ann. Sympos. Found. Comput. Sci., IEEE Computer Soc., New York (1979), pp. 397–410.

    Google Scholar 

  32. N. Ankeny, “The least quadratic nonresidue,” Ann. Math.,55, No. 2, 65–72 (1952).

    Google Scholar 

  33. L. Bernstein, “The Jacobi-Perron algorithm. Its theory and application,” Lect. Notes Math,207, Springer-Verlag, Berlin-Heidelberg-New York (1971).

    Google Scholar 

  34. W. A. Blankinship, “A new version of the Euclidean algorithm,” Am. Math. Mon.,70, 742–745 (1963).

    Google Scholar 

  35. J. Bond, “Calculating the general solution of a linear Diophantine equation,” Am. Math. Mon.,74, 955–957 (1976).

    Google Scholar 

  36. G. Bradly, “Algorithm and bound for the greatest common divisor of n integers,” Commun ACM,13, No. 7, 433–436 (1970).

    Google Scholar 

  37. A. J. Brentjes, “A two-dimensional continued fraction algorithm for best approximations with an application in cubic number fields,” J. Reine Angew. Math.,326, 18–44 (1981).

    Google Scholar 

  38. J. W. S. Cassels, “Bounds for the least solutions of homogeneous quadratic equations,” Proc. Camb. Philos. Soc.,51, Part 2, 262–264 (1955).

    Google Scholar 

  39. J. W. S. Cassels, Rational Quadratic Forms, Academic Press, London (1978).

    Google Scholar 

  40. A. Cobham, “The intrinsic computational complexity of functions,” Proc. 1964 Int. Congr. Logic, Amsterdam (1965), pp. 24–30.

  41. J. L. Donaldson, “Minkowski reduction of integral matrices,” Math. Comput.,63, No. 145, 201–216 (1979).

    Google Scholar 

  42. M. A. Frumkin, “Polynomial time algorithm in the theory of linear Diophantine equations,” in: Lect. Notes Comput. Sci.,56, Springer-Verlag, Berlin-Heidelberg-New York (1977), pp. 386–392.

    Google Scholar 

  43. M. R. Garey and D. S. Johnson, Computers and Intractability, W. H. Freeman, San Francisco (1979).

    Google Scholar 

  44. W. Jurkat, W. Kratz, and A. Peyerimhoff, “Explicit representation of Dirichlet approximations,” Math. Ann.,228, No. 1, 11–25 (1977).

    Google Scholar 

  45. W. Jurkat, W. Kratz, and A. Peyerimhoff, “On best two-dimensional Dirichlet-approximations and their algorithmic calculation,” Math. Ann.,244, No. 1, 1–33 (1979).

    Google Scholar 

  46. R. Kannan, “A polynomial algorithm for the two-variable integer-programming problems,” Tech. Rept. 348, Dept. Oper. Res., Cornell Univ., Ithaca, NY (1977).

    Google Scholar 

  47. R. Kannan and A. Bachem, “Polynomial algorithms for computing the Smith and Hermite forms of an integer matrix,” SIAM J. Comput.,8, No. 4, 499–507 (1979).

    Google Scholar 

  48. D. E. Knuth, “The analysis of algorithms,” Actes Congres Int. Math. (Nice, France, 1970), Vol. 3, Gauthier-Villars, Paris (1971), pp. 269–274.

    Google Scholar 

  49. J. C. Lagarias, “Succinct certificates for the solvability of binary quadratic Diophantine equations (extended abstract),” 20th Ann. Sympos. Found. Comput. Sci., IEEE Computer Soc., New York (1979), pp. 47–54.

    Google Scholar 

  50. J. C. Lagarias, “On the computational complexity of determining the solvability or unsolvability of the equation X2- DY2=−1,” Trans. Am. Math. Soc.,260, No. 2, 485–508 (1980).

    Google Scholar 

  51. D. H. Lehmer, “Euclid's algorithm for large numbers,” Am. Math. Mo.,45, 227–233 (1938).

    Google Scholar 

  52. H. W. Lenstra, Jr., “Integer programming with a fixed number of variables,” Rept. 81-03, Univ. of Amsterdam (1981).

  53. G. L. Miller, “Riemann's hypothesis and tests for primality,” J. Comput. Syst. Sci.,13, 300–317 (1976).

    Google Scholar 

  54. S. Morito and H. M. Salkin, “Using the Blankinship algorithm to find the general solution of a linear Diophantine equation,” Acta Inf.,13, No. 4, 379–382 (1980).

    Google Scholar 

  55. M. Pahst and H. Zassenhaus, “An effective number-geometric method of computing the fundamental units of an algebraic number field,” Math. Comput.,31, No. 139, 754–770 (1977).

    Google Scholar 

  56. V. Pratt, “Every prime has a succinct certificate,” SIAM J. Comput.,4, 214–220 (1975).

    Google Scholar 

  57. A. Schönhage, “Schnelle Berechnung von Kettenbruchentwicklungen,” Acta Inf.,1, 133–144 (1971).

    Google Scholar 

  58. F. Schweiger, “The metrical theory of Jacobi-Perron algorithm,” Lect. Notes Math.,334, Springer-Verlag, Berlin-Heidelberg-New York (1973).

    Google Scholar 

  59. F. Schweiger, “Approximation properties of Jacobi's algorithm,” Monatsh. Math.,81, No. 1, 59–61 (1976).

    Google Scholar 

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Authors
  1. M. A. Frumkin
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 118, pp. 188–210, 1982.

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Frumkin, M.A. Complexity questions in number theory. J Math Sci 29, 1502–1517 (1985). https://doi.org/10.1007/BF02104748

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