Mathematics of Computation

Author(s): Richard P. Brent; Samuli Larvala; Paul Zimmermann.
Journal: Math. Comp. 74 (2005), 1001-1002.
MSC (2000): Primary 11B83; Secondary 11-04, 11N35, 11R09, 11T06, 11Y55, 12-04
Posted: May 25, 2004
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Abstract: The only primitive trinomials of degree

over $\operatorname{GF}(2)$ are $x^{6972593} + x^{3037958} + 1$ and its reciprocal.

1.: R. P. Brent, Search for primitive trinomials, http://www.comlab.ox.ac.uk/oucl/work/ 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 -nomials , 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
6.: V. Shoup, NTL: A library for doing number theory (version 5.3.1), http://www.shoup.net/ ntl/.
7.: G. Woltman et al., GIMPS, The Great Internet Mersenne Prime Search, http://www. mersenne.org/.

Richard P. Brent
Affiliation: Oxford University Computing Laboratory, Oxford OX1 3QD, United Kingdom
Email: trinomials@rpbrent.co.uk

Samuli Larvala
Affiliation: Helsinki University of Technology, Espoo, Finland
Email: slarvala@cc.hut.fi

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