Revista Integración, temas de matemáticas.
Vol. 33 No. 1 (2015): Revista Integración, temas de matemáticas
Research and Innovation Articles

From Midy numbers to primality

John H. Castillo
Universidad de Nariño
Gilberto García-Pulgarín
Universidad de Antioquia
Juan Miguel Velásquez Soto
Universidad del Valle

Published 2015-05-21

Keywords

  • Prime numbers,
  • strong pseudoprimality,
  • Midy’s numbers,
  • Pocklington’s Theorem

How to Cite

Castillo, J. H., García-Pulgarín, G., & Velásquez Soto, J. M. (2015). From Midy numbers to primality. Revista Integración, Temas De matemáticas, 33(1), 1–10. https://doi.org/10.18273/revint.v33n1-2015001

Abstract

We define the concept of q-pseudoprime to base b, which extends the idea of strong pseudoprime to base b. We stablish a new test of primality that refines the Pocklinton’s Theorem using some properties of the Midy numbers.

To cite this article: J.H. Castillo, G. García-Pulgarín, J.M. Velásquez-Soto, De los números de Midy a la primalidad, Rev. Integr. Temas Mat. 33 (2015), no. 1, 1-10.

Downloads

Download data is not yet available.

References

  1. Adleman L.M., Pomerance C. and Rumely R.S., “On distinguishing prime numbers from composite numbers”, Ann. of Math. (2) 117 (1983), no. 1, 173–206.
  2. Agrawal M., Kayal N. and Saxena N., “PRIMES is in P”, Ann. of Math. (2) 160 (2004), no. 2, 781–793.
  3. Berrizbeitia P., “Sharpening PRIMES is in P for a large family of numbers”, Math. Comp. 74 (2005), no. 252, 2043–2059.
  4. Brillhart J. and Selfridge J.L., “Some factorizations of 2n ± 1 and related results”, Math. Comp. 21 (1967), 87-96; corrigendum, ibid. 21 (1967), 751.
  5. Castillo J.H., García-Pulgarín G. and Velásquez-Soto J.M., “Structure of associated sets to Midy’s Property”, Mat. Enseñ. Univ. 20 (2012), no. 1, 21–28.
  6. Cheng Q., “Primality proving via one round in ECPP and one iteration in AKS”, J. Cryptology. 20 (2007), no. 3, 375–387.
  7. Crandall R. and Pomerance C., Prime numbers. A computational perspective, Springer, New York, 2005.
  8. García-Pulgarín G. and Giraldo H., “Characterizations of Midy’s property”, Integers 9 (2009), 191–197.
  9. Gauss C.F., “Disquisitiones arithmeticae”, in Colección Enrique Pérez Arbeláez , Academia Colombiana de Ciencias Exactas, Físicas y Naturales, Translated from the Latin by Hugo Barrantes Campos, Michael Josephy and Ángel Ruiz Zúñiga, with a preface by Ruiz Zúñiga, 10 (1995).
  10. Lenstra H.W. Jr. and Pomerance C., “Primality testing with gaussian periods”, https://www.math.dartmouth.edu/ carlp/aks041411.pdf, consultado el día 22 de abril de
  11. , unpublished.
  12. Motose K., “On values of cyclotomic polynomials. II”, Math. J. Okayama Univ. 37 (1995), 27–36.
  13. Nathanson M.B., Elementary methods in number theory, Springer-Verlag, New York, 2000.
  14. Shevelev V., Castillo J.H., García-Pulgarín G. and Velásquez-Soto J.M., “Overpseudoprimes, and Mersenne and Fermat Numbers as Primover Numbers”, J. Integer Seq. 15
  15. (2012), no. 7, 1-10.
  16. Zhang Z., “Notes on some new kinds of pseudoprimes”, Math. Comp. 75 (2006), no. 253, 451–460.