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A primitive trinomial of degree 6972593

Authors: Richard P. Brent, Samuli Larvala and Paul Zimmermann
Journal: Math. Comp. 74 (2005), 1001-1002
MSC (2000): Primary 11B83; Secondary 11-04, 11N35, 11R09, 11T06, 11Y55, 12-04
Published electronically: May 25, 2004
MathSciNet review: 2114660
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Abstract | References | Similar Articles | Additional Information

Abstract: The only primitive trinomials of degree $6972593$ over $\operatorname{GF}(2)$are $x^{6972593} + x^{3037958} + 1$ and its reciprocal.

References [Enhancements On Off] (What's this?)

  • 1. R. P. Brent, Search for primitive trinomials, richard.brent/trinom.html.
  • 2. R. P. Brent, S. Larvala and P. Zimmermann, A fast algorithm for testing reducibility of trinomials mod 2 and some new primitive trinomials of degree 3021377, Math. Comp. 72 (2003), 1443-1452.
  • 3. R. P. Brent and P. Zimmermann, Algorithms for finding almost irreducible and almost primitive trinomials, in Primes and Misdemeanours: Lectures in Honour of the Sixtieth Birthday of Hugh Cowie Williams, Fields Institute, Toronto, 2004, to appear.
  • 4. T. Kumada, H. Leeb, Y. Kurita and M. Matsumoto, New primitive $t$-nomials $(t = 3$, $5)$ over $\operatorname{GF}(2)$whose degree is a Mersenne exponent, Math. Comp. 69 (2000), 811-814; Corrigenda: ibid 71 (2002), 1337-1338. MR 2000i:11183; MR 2003c:11153
  • 5. T. J. Nicely, Enumeration to $10^{14}$ of the twin primes and Brun's constant, Virginia Journal of Science 46 (1995), 195-204. MR 97e:11014
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Additional Information

Richard P. Brent
Affiliation: Oxford University Computing Laboratory, Oxford OX1 3QD, United Kingdom

Samuli Larvala
Affiliation: Helsinki University of Technology, Espoo, Finland

Paul Zimmermann
Affiliation: LORIA/INRIA Lorraine, 615 rue du jardin botanique, BP 101, F-54602 Villers-lès-Nancy, France

Keywords: Irreducible trinomials, primitive trinomials, Mersenne numbers
Received by editor(s): August 26, 2003
Received by editor(s) in revised form: October 6, 2003
Published electronically: May 25, 2004
Article copyright: © Copyright 2004 American Mathematical Society