Sequences of consecutive smooth polynomials over a finite field
HTML articles powered by AMS MathViewer
- by Ariane Masuda and Daniel Panario PDF
- Proc. Amer. Math. Soc. 135 (2007), 1271-1277 Request permission
Abstract:
Given $\varepsilon > 0$, we show that there are infinitely many sequences of consecutive $\varepsilon n$-smooth polynomials over a finite field. The number of polynomials in each sequence is approximately $\ln \ln \ln n$.References
- Antal Balog and Trevor D. Wooley, On strings of consecutive integers with no large prime factors, J. Austral. Math. Soc. Ser. A 64 (1998), no. 2, 266–276. MR 1619809, DOI 10.1017/S1446788700001750
- Gove Effinger, A Goldbach theorem for polynomials of low degree over odd finite fields, Acta Arith. 42 (1983), no. 4, 329–365. MR 736718, DOI 10.4064/aa-42-4-329-365
- Gove Effinger, A Goldbach $3$-primes theorem for polynomials of low degree over finite fields of characteristic $2$, J. Number Theory 29 (1988), no. 3, 345–363. MR 955958, DOI 10.1016/0022-314X(88)90111-4
- Gove W. Effinger, Kenneth H. Hicks, and Gary L. Mullen, Twin irreducible polynomials over finite fields, Finite fields with applications to coding theory, cryptography and related areas (Oaxaca, 2001) Springer, Berlin, 2002, pp. 94–111. MR 1995330
- R. B. Eggleton and J. L. Selfridge, Consecutive integers with no large prime factors, J. Austral. Math. Soc. Ser. A 22 (1976), no. 1, 1–11. MR 439778, DOI 10.1017/s1446788700013318
- Theodoulos Garefalakis and Daniel Panario, The index calculus method using non-smooth polynomials, Math. Comp. 70 (2001), no. 235, 1253–1264. MR 1826581, DOI 10.1090/S0025-5718-01-01298-4
- Theodoulos Garefalakis and Daniel Panario, Polynomials over finite fields free from large and small degree irreducible factors, J. Algorithms 44 (2002), no. 1, 98–120. Analysis of algorithms. MR 1932679, DOI 10.1016/S0196-6774(02)00207-9
- C. J. Hall, $L$-functions of Twisted Curves, Ph.D. Thesis, Princeton University, 2003.
- G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909
- Kenneth H. Hicks, Gary L. Mullen, and Ikuro Sato, Distribution of irreducible polynomials over $\mathbf F_2$, Finite fields with applications to coding theory, cryptography and related areas (Oaxaca, 2001) Springer, Berlin, 2002, pp. 177–186. MR 1995335
- Heinrich Kornblum and E. Landau, Über die Primfunktionen in einer arithmetischen Progression, Math. Z. 5 (1919), no. 1-2, 100–111 (German). MR 1544375, DOI 10.1007/BF01203156
- R. Lovorn, Rigorous, Subexponential Algorithms for Discrete Logarithms over Finite Fields, Ph.D. Thesis, University of Georgia, 1992.
- A. M. Odlyzko, Discrete logarithms in finite fields and their cryptographic significance, Advances in cryptology (Paris, 1984) Lecture Notes in Comput. Sci., vol. 209, Springer, Berlin, 1985, pp. 224–314. MR 825593, DOI 10.1007/3-540-39757-4_{2}0
- A. M. Odlyzko, Discrete logarithms and smooth polynomials, Finite fields: theory, applications, and algorithms (Las Vegas, NV, 1993) Contemp. Math., vol. 168, Amer. Math. Soc., Providence, RI, 1994, pp. 269–278. MR 1291435, DOI 10.1090/conm/168/01706
- Gérald Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, 2nd ed., Cours Spécialisés [Specialized Courses], vol. 1, Société Mathématique de France, Paris, 1995 (French). MR 1366197
Additional Information
- Ariane Masuda
- Affiliation: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6
- MR Author ID: 791815
- Email: ariane@math.carleton.ca
- Daniel Panario
- Affiliation: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6
- Email: daniel@math.carleton.ca
- Received by editor(s): January 19, 2005
- Received by editor(s) in revised form: December 14, 2005
- Published electronically: November 13, 2006
- Communicated by: Wen-Ching Winnie Li
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 1271-1277
- MSC (2000): Primary 11T06, 11T99
- DOI: https://doi.org/10.1090/S0002-9939-06-08611-4
- MathSciNet review: 2276634