Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T22:22:07.023Z Has data issue: false hasContentIssue false
  • English
  • Français

INFINITUDE OF ELLIPTIC CARMICHAEL NUMBERS

Part of: Computational number theory Elementary number theory Multiplicative number theory

Published online by Cambridge University Press:  25 April 2012

AARON EKSTROM ,
CARL POMERANCE  and
DINESH S. THAKUR
AARON EKSTROM
Affiliation:
8505 E. Ocotillo Dr., Tucson AZ 85750, USA (email: DoctorX198@gmail.com)
CARL POMERANCE*
Affiliation:
Mathematics Department, Dartmouth College, Hanover, NH 03755, USA (email: carl.pomerance@dartmouth.edu)
DINESH S. THAKUR
Affiliation:
Mathematics Department, University of Arizona, Tucson, AZ 85721, USA (email: thakur@math.arizona.edu)
*
✉For correspondence; e-mail: carl.pomerance@dartmouth.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In 1987, Gordon gave an integer primality condition similar to the familiar test based on Fermat’s little theorem, but based instead on the arithmetic of elliptic curves with complex multiplication. We prove the existence of infinitely many composite numbers simultaneously passing all elliptic curve primality tests assuming a weak form of a standard conjecture on the bound on the least prime in (special) arithmetic progressions. Our results are somewhat more general than both the 1999 dissertation of the first author (written under the direction of the third author) and a 2010 paper on Carmichael numbers in a residue class written by Banks and the second author.

Keywords

probable primespseudo-primesprimality testselliptic curvescomplex multiplication

MSC classification

Secondary: 11A51: Factorization; primality 11Y11: Primality 11N13: Primes in progressions 11N36: Applications of sieve methods 11A07: Congruences; primitive roots; residue systems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 92 , Issue 1 , February 2012 , pp. 45 - 60
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

The second author was supported in part by NSF grant DMS-1001180. The third author was supported in part by NSA grant H98230-10-1-0200.

References

[1]Alford, W. R., Granville, A. and Pomerance, C., ‘There are infinitely many Carmichael numbers’, Ann. of Math. (2) 139 (1994), 703–722.CrossRefGoogle Scholar
[2]Alford, W. R., Granville, A. and Pomerance, C., ‘On the difficulty of finding reliable witnesses’, in: Algorithmic Number Theory, Ithaca, NY, 1994, Lecture Notes in Computer Science, 877 (Springer, Berlin, 1994), pp. 1–16.Google Scholar
[3]Atkin, A. O. L. and Morain, F., ‘Elliptic curves and primality proving’, Math. Comp. 61 (1993), 29–68.CrossRefGoogle Scholar
[4]Bach, E. and Shallit, J., Algorithmic Number Theory, Foundations of Computing (MIT Press, Cambridge, MA, 1996).Google Scholar
[5]Baker, R. C. and Harman, G., ‘Shifted primes without large prime factors’, Acta Arith. 83 (1998), 331–361.CrossRefGoogle Scholar
[6]Balasubramanian, R. and Murty, M. R., ‘Elliptic pseudoprimes. II’, in: Séminaire de théorie des nombres, Paris 1988-1989, Progress in Mathematics, 91 (Birkhäuser, Boston, 1990), pp. 13–25.Google Scholar
[7]Banks, W. D. and Pomerance, C., ‘On Carmichael numbers in arithmetic progressions’, J. Aust. Math. Soc. 28 (2010), 313–321.CrossRefGoogle Scholar
[8]Carmichael, R. D., ‘Note on a new number theory function’, Bull. Amer. Math. Soc. 16 (1910), 232–238.CrossRefGoogle Scholar
[9]Carmichael, R. D., ‘On composite numbers P which satisfy the Fermat congruence a P−1≡1 mod P’, Amer. Math. Monthly 19(156) (1956), 22–27.Google Scholar
[10]Chernick, J., ‘On Fermat’s simple theorem’, Bull. Amer. Math. Soc. 45 (1939), 269–274.CrossRefGoogle Scholar
[11]Cojocaru, A., Luca, F. and Shparlinski, I. E., ‘Pseudoprime reductions of elliptic curves’, Math. Proc. Cambridge Philos. Soc. 146 (2009), 513–522.CrossRefGoogle Scholar
[12]Crandall, R. and Pomerance, C., Prime Numbers: A Computational Perspective, 2nd edn (Springer, New York, 2005).Google Scholar
[13]David, C. and Wu, J., Pseudoprime reductions of elliptic curves. arXiv:1005.3871.Google Scholar
[14]Deuring, M., ‘Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins. IV’, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. IIa 1957 (1957), 55–80.Google Scholar
[15]Ekstrom, A., ‘On the infinitude of elliptic Carmichael numbers’, PhD Thesis, University of Arizona, 1999.Google Scholar
[16]Erdős, P., ‘On some applications of Brun’s method’, Acta Univ. Szeged. Sect. Sci. Math. 3 (1949), 57–63.Google Scholar
[17]Goldwasser, S. and Kilian, J., ‘Almost all primes can be quickly certified’, in: Proc. 18th Annual ACM Sympos. on Theory of Computing, STOC, Berkeley 1986 (Association for Computing Machinery, New York, 1986), pp. 316–329.Google Scholar
[18]Gordon, D. M., ‘Pseudoprimes on elliptic curves’, in: Number Theory, Proc. Internat. Number Theory Conf., Laval 1987, (eds. De Koninck, J. M. and Levesque, C.) (de Gruyter, New York, 1989), pp. 291–305.Google Scholar
[19]Gordon, D. M., ‘On the number of elliptic pseudoprimes’, Math. Comp. 52 (1989), 231–245.CrossRefGoogle Scholar
[20]Gordon, D. M. and Pomerance, C., ‘The distribution of Lucas and elliptic pseudoprimes’, Math. Comp. 57 (1991), 825–838.CrossRefGoogle Scholar
[21]Granville, A., ‘Least primes in arithmetic progressions’, in: Number Theory, Proc. Internat. Number Theory Conf., Laval 1987 (eds. De Koninck, J. M. and Levesque, C.) (de Gruyter, New York, 1989), pp. 306–321.Google Scholar
[22]Harman, G., ‘Watt’s mean value theorem and Carmichael numbers’, Int. J. Number Theory 4 (2008), 241–248.CrossRefGoogle Scholar
[23]Heath-Brown, D. R., ‘Almost-primes in arithmetic progressions and short intervals’, Math. Proc. Cambridge Philos. Soc. 83 (1978), 357–375.CrossRefGoogle Scholar
[24]Heath-Brown, D. R., ‘Zero-free regions for Dirichlet L-functions and the least prime in an arithmetic progression’, Proc. Lond. Math. Soc. 64 (1992), 265–338.CrossRefGoogle Scholar
[25]Ito, H., ‘On elliptic pseudoprimes’, Mem. College Ed. Akita Univ. Natur. Sci. 46 (1994), 1–7.Google Scholar
[26]Korselt, A., ‘Problème chinois’, L’intermédiaire des Mathématiciens 6 (1899), 142–143.Google Scholar
[27]Lehmer, D. H., ‘Strong Carmichael numbers’, J. Aust. Math. Soc. Ser. A 21 (1976), 508–510.CrossRefGoogle Scholar
[28]Lenstra, H. W. Jr, ‘Elliptic curves and number-theoretic algorithms’, Proc. ICM, Berkeley, California, USA, 1986, (1986), pp. 99–120.Google Scholar
[29]Lenstra, H. W. Jr, ‘Factoring integers with elliptic curves’, Ann. of Math. (2) 126 (1987), 649–673.CrossRefGoogle Scholar
[30]Linnik, U. V., ‘On the least prime in an arithmetic progression II’, Rec. Math. [Mat. Sb.] 15 (1944), 347–368.Google Scholar
[31]Matomäki, K., ‘Carmichael numbers in arithmetic progressions’, Preprint, 2011.Google Scholar
[32]McCurley, K. S., ‘The least r-free number in an arithmetic progression’, Trans. Amer. Math. Soc. 293 (1986), 467–475.Google Scholar
[33]Miyamoto, I. and Murty, M. R., ‘Elliptic pseudoprimes’, Math. Comp. 53 (1989), 415–430.CrossRefGoogle Scholar
[34]Montgomery, H. L. and Vaughan, R. C., ‘The large sieve’, Mathematika 20 (1973), 119–134.CrossRefGoogle Scholar
[35]Montgomery, H. L. and Vaughan, R. C., ‘Multiplicative number theory I’, in: Classical Theory (Cambridge University Press, Cambridge, 2007).Google Scholar
[36]Müller, S., ‘On the existence and nonexistence of elliptic pseudoprimes’, Math. Comp. 79 (2010), 1171–1190.CrossRefGoogle Scholar
[37]Pomerance, C., ‘A note on the least prime in an arithmetic progression’, J. Number Theory 12 (1080), 218–223.CrossRefGoogle Scholar
[38]Pomerance, C., ‘On the distribution of pseudoprimes’, Math. Comp. 37 (1981), 587–593.CrossRefGoogle Scholar
[39]Pomerance, C., Selfridge, J. L. and Wagstaff, S. S. Jr., ‘The pseudoprimes to 25⋅109’, Math. Comp. 35 (1980), 1003–1026.Google Scholar
[40]van der Poorten, A. J. and Rotkiewicz, A., ‘On strong pseudoprimes in arithmetic progressions’, J. Aust. Math. Soc. Ser. A 29 (1980), 316–321.CrossRefGoogle Scholar
[41]Silverman, J. H., ‘Elliptic Carmichael numbers and elliptic Korselt criteria’, arXiv:1108.3830.Google Scholar
[42]Silverman, J. H., The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106 (Springer, New York, 1986).CrossRefGoogle Scholar
[43]Uchiyama, S., ‘An application of the large sieve’, Proc. Japan Acad. 48 (1972), 67–69.Google Scholar
[44]Wagstaff, S. S. Jr, ‘Greatest of the least primes in arithmetic progressions having a given modulus’, Math. Comp. 33 (1979), 1073–1080.CrossRefGoogle Scholar
[45]Xylouris, T., ‘Über die Linniksche Konstante’, arXiv:0906.2749.Google Scholar