Siberian Mathematical Journal

, Volume 54, Issue 3, pp 501–516 | Cite as

Solvability of cubic equations in p-ADIC integers (p > 3)

  • F. M. Mukhamedov
  • B. A. Omirov
  • M. Kh. Saburov
  • K. K. Masutova


We give a criterion for the existence of solutions to an equation of the form x 3 + ax = b, where a, b ∈ ℚ p , in p-adic integers for p > 3. Moreover, in the case when the equation x 3 + ax = b is solvable, we give necessary and sufficient recurrent conditions on a p-adic number x ∈ ℤ* p under which x is a solution to the equation.


cubic equation p-adic number solution algorithm 


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  1. 1.
    Aref’eva L. Ya., Dragovich B., Frampton P. H., and Volovich I. V., “The wave function of the universe and p-adic gravity,” Internat. J. Modern Phys. A, 6, 4341–4358 (1991).MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Beltrametti E. and Cassinelli G., “Quantum mechanics and p-adic numbers,” Found. Phys., 2, 1–7 (1972).MathSciNetCrossRefGoogle Scholar
  3. 3.
    Freund P. G. O. and Witten E., “Adelic string amplitudes,” Phys. Lett. B, 199, No. 2, 191–194 (1987).MathSciNetCrossRefGoogle Scholar
  4. 4.
    Khrennikov A. Yu., “p-Adic quantum mechanics with p-adic valued functions,” J. Math. Phys., 32, No. 4, 932–936 (1991).MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Khrennikov A. Yu., p-Adic Valued Distributions in Mathematical Physics, Kluwer, Dordrecht (1994).zbMATHCrossRefGoogle Scholar
  6. 6.
    Manin Yu., “New dimensions in geometry,” Lect. Notes Math., 1111, 59–101 (1985).MathSciNetCrossRefGoogle Scholar
  7. 7.
    Marinari E. and Parisi G., “On the p-adic five-point function,” Phys. Lett., 203, 52–56 (1988).MathSciNetGoogle Scholar
  8. 8.
    van der Blij F. and Monna A. F., “Models of space and time in elementary physics,” J. Math. Anal. Appl., 22, 537–545 (1968).zbMATHCrossRefGoogle Scholar
  9. 9.
    Mukhamedov F. M. and Rozikov U. A., “On Gibbs measures of p-adic Potts model on the Cayley tree,” Indag. Math. (N.S.), 15, No. 1, 85–100 (2004).MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Mukhamedov F. M. and Rozikov U. A., “On inhomogeneous p-adic Potts model on a Cayley tree,” Infin. Dimens. Anal. Quantum Probab. Relat. Top., 8, No. 2, 277–290 (2005).MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Volovich I. V., “p-Adic string,” Classical Quantum Gravity, 4, No. 4, 83–87 (1987).MathSciNetCrossRefGoogle Scholar
  12. 12.
    Vladimirov V. S., Volovich I. V., and Zelenov E. I., p-Adic Analysis and Mathematical Physics, World Scientific, Singapore (1994).CrossRefGoogle Scholar
  13. 13.
    Lang S., Algebraic Number Theory, Springer-Verlag, New York (1994).zbMATHCrossRefGoogle Scholar
  14. 14.
    Neukirch J., Algebraic Number Theory, Springer-Verlag, Berlin (1999).zbMATHCrossRefGoogle Scholar
  15. 15.
    Serre J.-P., Local Fields, Springer-Verlag, New York (1979).zbMATHCrossRefGoogle Scholar
  16. 16.
    Gouvea F. Q., P-Adic Numbers. An Introduction, Springer-Verlag, Berlin (1997).zbMATHCrossRefGoogle Scholar
  17. 17.
    Koblitz N., p-Adic Numbers, p-Adic Analysis, and Zeta Functions, Springer-Verlag, New York (1984).CrossRefGoogle Scholar
  18. 18.
    Schikhof W. H., Ultrametric Calculus: an Introduction to p-Adic Analysis, Cambridge Univ. Press, Cambridge (1984).Google Scholar
  19. 19.
    Avendaño M., Ibrahim A., Rojas J. M., and Rusek K., “Near NP-completeness for detecting p-adic rational roots in one variable,” arXiv:1001.4252.Google Scholar
  20. 20.
    Avendaño M., Ibrahim A., Rojas J. M., and Rusek K., “Faster p-adic feasibility for certain multivariate sparse polynomials,” arXiv:1010.5310.Google Scholar
  21. 21.
    Mukhamedov F. and Saburov M., “On equation x q = a over ℚp,” J. Number Theory, 13, 55–58 (2013).MathSciNetCrossRefGoogle Scholar
  22. 22.
    Casas J. M., Omirov B. A., and Rozikov U. A., “Solvability criteria for the equation x q = a in the field of p-adic numbers,” arXiv:1102.2156.Google Scholar
  23. 23.
    Ayupov Sh. A. and Kurbanbaev T. K., “The classification of 4-dimensional p-adic filiform Leibniz algebras,” TWMS J. Pure Appl. Math., 1, No. 2, 155–162 (2010).MathSciNetzbMATHGoogle Scholar
  24. 24.
    Khudoyberdiyev A. Kh., Kurbanbaev T. K., and Omirov B. A., “Classification of three-dimensional solvable p-adic Leibniz algebras,” P-Adic Numbers Ultrametric Anal. Appl., 2, 207–221 (2010).MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Mukhamedov F., Omirov B., and Saburov M., “On cubic equations over p-adic field,” arXiv 1204.1743.Google Scholar
  26. 26.
    Khudoyberdiyev A. Kh., Kurbanbaev T. K., and Masutova K. K., “On the solutions of the equation x 3 + ax = b in ℤ3* with coefficients from ℚ3,” arXiv: 1110.1010.Google Scholar
  27. 27.
    Serre J.-P., “On a theorem of Jordan,” Bull. Amer. Math. Soc., 40, 429–440 (2003).MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Sun Z. H., “On the theory of cubic residues and nonresidues,” Acta Arith., 84, 291–335 (1998).MathSciNetzbMATHGoogle Scholar
  29. 29.
    Sun Z. H., “Cubic and quartic congruences modulo a prime,” J. Number Theory, 102, 41–89 (2003).MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Sun Z. H., “Cubic residues and binary quadratic forms,” J. Number Theory, 124, 62–104 (2007).MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Borevich Z. I. and Shafarevich I. R., Number Theory, Academic Press, New York (1966).Google Scholar
  32. 32.
    Rosen K. H., Elementary Number Theory and Its Applications, Addison Wesley, Pearson (2011).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  • F. M. Mukhamedov
    • 1
  • B. A. Omirov
    • 2
  • M. Kh. Saburov
    • 1
  • K. K. Masutova
    • 2
  1. 1.International Islamic University MalaysiaKuantan, PahangMalaysia
  2. 2.Institute of Mathematics at the National University of UzbekistanTashkentUzbekistan

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