Skip to main content

Solved and Unsolved Problems on Pseudoprime Numbers and Their Generalizations

  • Chapter
Applications of Fibonacci Numbers
  • 459 Accesses

Abstract

Following recent papers [1], [5], [22], [23] a composite n is called a pseudoprime to base \(b if {{b}^{{n - 1}}} \equiv 1 \bmod n. \)

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alford, W.R., Granville, A. and Pomerance C. “There are infinitely many Carmichael numbers.” Ann. of Math., Vol. 140 (1994): pp. 703–722.

    Article  MathSciNet  Google Scholar 

  2. Baillie R. and Wagstaff Samuel S. “Lucas pseudoprimes.” Math. Comp., Vol. 35 (1980): pp. 1391–1417.

    Article  MathSciNet  MATH  Google Scholar 

  3. Benkoski, S.J. Review of On the congruence \({{{\text{2}}}^{{n - k}}} \equiv \left( {\bmod n} \right). \). MR87e: 11005.

    Google Scholar 

  4. Cipolla, M. “Sui numeri composti P che verificaro la congruenza di Fermat \({{a}^{{P - 1}}} \equiv 1\left( {\bmod P} \right). \).” Annali di Matematica (3), Vol. 9 (1904): pp. 139–160.

    Article  Google Scholar 

  5. Conway, J.H., Guy, R.K., Schneeberger, W.A. and Sloane, N.J.A. “The primary pretenders.” Acta Arith., Vol. 78 (1997): pp. 307–313.

    MathSciNet  Google Scholar 

  6. Dickson, L.E. History of the Theory of Numbers. Carnegie Institute, Washington, 1919, 1920, 1923; reprinted Stechert, New York, 1934; Chelsea, New York, 1952, 1960, Vol. I, Chap. III.

    MATH  Google Scholar 

  7. Erdös, P., Kiss, P. and Sárközy, A. “Lower bound for the counting function.” Math. Comp., Vol. 51 (1988): pp. 315–323.

    MathSciNet  MATH  Google Scholar 

  8. Fehér, J. and Kiss, P. “Note on super pseudoprime numbers.” Ann. Univ. Sci. Budapest, Eötvös Sect. Math., Vol. 26 (1983): pp. 157–159.

    MATH  Google Scholar 

  9. Grantham, J.F. Frobenius pseudoprimes. A dissertation submitted to the Graduate Faculty of the University of Georgia, Athens GA 1997.

    Google Scholar 

  10. Granville, A. The prime k-tuplets conjecture implies that there are arbitrarily long arithmetic progressions of Carmichael numbers, (written communication).

    Google Scholar 

  11. Guy, R.K. Unsolved Problems in Number Theory. Springer-Verlag, New York-Heidelberg, XVI + 285, p.p. 1994.

    Google Scholar 

  12. Kiss, P. and Phong, B.M. “On a problem of A. Rotkiewicz.” Math. Comp., Vol. 48 (1987): pp. 751–755.

    Article  MathSciNet  MATH  Google Scholar 

  13. Korselt, A. “Probléme chinois.” L’intermédaire des mathématiciens, Vol. 6 (1899): pp. 142–143.

    Google Scholar 

  14. Lehmer E. “On the infinitude of Fibonacci pseudoprimes.” The Fibonacci Quarterly, Vol. 2 (1964): pp. 229–230.

    MATH  Google Scholar 

  15. Mahnke, D. “Leibniz and der Suche nach einer allgemeinem Primzahlgleichung.” Bibliotheca Math., Vol. 13 (1913): pp. 29–61.

    Google Scholar 

  16. Ma¸kowski, A. “Generalization of Morrow’s D numbers.” Simon Stevin, Vol. 36 (1962): p. 71.

    MathSciNet  Google Scholar 

  17. McDaniel, W.L. “The generalized pseudoprime congruence \({{a}^{{n - k}}} \equiv {{b}^{{n - k}}}\left( {\bmod n} \right). \).” C. R. Math. Rep. Acad. Sci. Canada, Vol. 9 (1987): pp. 143–148.

    MathSciNet  MATH  Google Scholar 

  18. McDaniel, W.L. “Some pseudoprimes and related numbers having special forms.” Math. Comp., Vol. 53 (1989): pp. 407–409.

    Article  MathSciNet  MATH  Google Scholar 

  19. McDaniel, W.L. “The existence of solutions of the generalized pseudoprime congruence \({{a}^{{f(n)}}} \equiv {{b}^{{f(n)}}}\left( {\bmod n} \right). \).” Colloq. Math., Vol. 49 (1990): pp. 177–190.

    MathSciNet  Google Scholar 

  20. Morrow, D.C. “Some properties of D numbers.” Amer. Math. Monthly, Vol. 58 (1951): pp. 329–330.

    Article  MathSciNet  MATH  Google Scholar 

  21. Phong, B.M. “Generalized solutions of Rotkiewicz’s problem.” Matematikai Lapok, Vol. 34(1–3) (1987): pp. 109–119.

    MATH  Google Scholar 

  22. Pomerance, C. “A new lower bound for the pseudoprimes counting function.” Illinois J. Math., Vol. 26 (1982): pp. 4–9; MR 83h: 1012.

    MathSciNet  MATH  Google Scholar 

  23. Pomerance, C., Selfridge, I.L. and Wagstaff, Samuel S. “The pseudoprimes to 25 · 109.” Math. Comp., Vol. 35 (1980): pp. 1003–1026.

    MathSciNet  MATH  Google Scholar 

  24. Ribenboim, P. The New Book of Prime Number Records, Springer-Verlag, New York-Berlin-Heidelberg, 1995.

    Google Scholar 

  25. Rotkiewicz, A. “Sur les nombres composés n qui divisent a n-1-b n-1.” Rend. Circ. Mat. Palermo (2), Vol. 8 (1959): pp. 115–116; MR. 23#A1579.

    Article  MathSciNet  MATH  Google Scholar 

  26. Rotkiewicz, A. “Sur quelques généralisations des nombres pseudopremiers.” Colloq. Math., Vol. 9 (1962): pp. 109–113.

    MathSciNet  MATH  Google Scholar 

  27. Rotkiewicz, A. “Sur les nombres premiers p et q tels que pq | 2pq-2.” Rend Circ. Mat. Palermo, Vol. 11 (1962): pp. 280–282.

    Article  MathSciNet  MATH  Google Scholar 

  28. Rotkiewicz, A. “Sur les nombres pseudopremiers de la forme ax + b.” C. R. Acad. Sci. Paris, Vol. 257 (1963): pp. 2601–2604.

    MathSciNet  MATH  Google Scholar 

  29. Rotkiewicz, A. “Sur progressions arithmétiques et gómétriques formés de trois nombres pseudopremiers distincts.” Acta Arith., Vol. 10 (1964): pp. 325–328.

    MathSciNet  MATH  Google Scholar 

  30. Rotkiewicz, A. “Sur les nombres pseudopremiers triangulaires.” Elem. Math., Vol. 19 (1964): pp. 82–83.

    MathSciNet  MATH  Google Scholar 

  31. Rotkiewicz, A. “Sur les nombres pseudopremiers pentagonaux.” Bull. Soc. Roy. Sci. Liége, Vol. 33 (1964): pp. 261–263.

    MathSciNet  MATH  Google Scholar 

  32. Rotkiewicz, A. “On the pseudoprimes of the form ax + b.” Proc. Combridge Phil. Soc., Vol. 63 (1963): pp. 389–392.

    Article  MathSciNet  Google Scholar 

  33. Rotkiewicz, A. “On the prime factors of the number 2p-1-1.” Glasgow Math. J., Vol. 9 (1968): pp. 83–86.

    Article  MathSciNet  MATH  Google Scholar 

  34. Rotkiewicz, A. “On arithmetical progressions formed by k different pseudoprimes.” Journal of Math. Sciences, Vol. 4 (1969): pp. 5–10.

    MathSciNet  Google Scholar 

  35. Rotkiewicz, A. “Pseudoprime Numbers and Their Generalizations.” Student Association of Faculty of Sciences, Univ. of Novi Sad, 1972, i + 169, pp.

    Google Scholar 

  36. Rotkiewicz, A. “The solution of W. Sierpiński’s problem.” Rend. Circ. Mat. Palermo, Vol. 28 (1979): pp. 62–64.

    Article  MathSciNet  MATH  Google Scholar 

  37. Rotkiewicz, A. “Arithmetical progressions formed from three different Euler pseudoprimes for the odd base a.” Rend. Circ. Mat. Palermo, Vol. 29 (1980): pp. 420–426.

    Article  MathSciNet  MATH  Google Scholar 

  38. Rotkiewicz, A. “On the congruence \({{2}^{{n - 2}}} \equiv 1\left( {\bmod n} \right). \).” Math. Comp., Vol. 43 (1984): pp. 271–272.

    MathSciNet  MATH  Google Scholar 

  39. Rotkiewicz, A. “Arithmetical progressions formed by k different pseudoprimes.” Rend. Circ. Mat. Palermo, Vol. 69 (1994).

    Google Scholar 

  40. Rotkiewicz, A. “There are infinitely many arithmetical progressions formed by three different Fibonacci pseudoprimes.” Applications of Fibonacci Numbers. Volume 7. Edited by G.E. Bergum, A.F. Horadam and A.N. Philippou. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998: pp. 327–332.

    Chapter  Google Scholar 

  41. Rotkiewicz, A. “Arithmetical progressions formed by Lucas pseudoprimes.” Number Theory, Proceedings Eger Conference. Edited by K. Györy, A. Pethö, V.T. Sós, Walter de Gruyter. Berlin-New York 1998: pp. 465–472.

    Google Scholar 

  42. Rotkiewicz, A. “Lucas and Lehmer pseudoprimes with negative Jacobi symbol.” (to appear in Acta Arith.).

    Google Scholar 

  43. Rotkiewicz, A. “Periodic sequences of pseudoprimes connected with Carmichael numbers and the least period of the function l x C.” (to appear in Acta Anth.).

    Google Scholar 

  44. Schinzel, A. “Sur les nombres composés n qui divisent a n-a.” Rend. Circ. Mat. Palermo (2), Vol. 7 (1958): pp. 37–41.

    Article  MathSciNet  MATH  Google Scholar 

  45. Schinzel, A. and Sierpiński, W. “Sur certaines hypothèses concernnt les nombres premiers.” Acta Arith., Vol. 4 (1958): pp. 185–208, Correction ibid. Vol. 5 (1958): p. 259.

    MATH  Google Scholar 

  46. Shen, Mok-King. “On the congruence \({{2}^{{n - 2}}} \equiv 1\left( {\bmod n} \right). \).” Math. Comp., Vol. 46 (1986): pp. 715–716.

    MathSciNet  MATH  Google Scholar 

  47. Sierpiński, W. “Remark on composite numbers m dividing a m-a (in Polish).” Wiadom. Mat, Vol. 4 (1961): pp. 183–184.

    MathSciNet  MATH  Google Scholar 

  48. Sierpiński, W. “Elementary Theory of Numbers.” Monografie Matematyczne, Vol. 42 PWN, Warsaw (1964), (second edition, North-Holland, Amsterdam, 1987).

    Google Scholar 

  49. Williams, H.C. “On numbers analogous to the Carmichael numbers.” Canad. Math. Bull., Vol. 20.1 (1977): pp. 133–143.

    Article  Google Scholar 

  50. Zhang, Ming-Zhi. “A note on the congruence \({{{\text{2}}}^{{n - 2}}} \equiv \bmod n \).” (Chinese and English summary), Sichuan Daxue Xuebao, Vol. 27 (1990): pp. 130–131.

    MathSciNet  MATH  Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Rotkiewicz, A. (1999). Solved and Unsolved Problems on Pseudoprime Numbers and Their Generalizations. In: Howard, F.T. (eds) Applications of Fibonacci Numbers. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4271-7_28

Download citation

  • DOI: https://doi.org/10.1007/978-94-011-4271-7_28

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-5851-3

  • Online ISBN: 978-94-011-4271-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics