## Section: New Results

### Exact arithmetic

Participants : Pierrick Gaudry, Guillaume Hanrot, Alexander Kruppa, Emmanuel Thomé, Paul Zimmermann.

Richard Brent and P. Zimmermann are collaborating on a book called “Modern Computer Arithmetic”. Three new versions (0.2.1 in March, 0.3 in June, and 0.4 in November) have been published in 2009, in the context of the Inria ANC associate team(http://www.loria.fr/~zimmerma/anc.html ) which started in 2008. Version 0.4 includes feedback from two anonymous reviewers for a commercial publisher.

Another common project with Richard Brent is the search for primitive trinomials over . While the paper corresponding to degrees 24036583, 25964951, 30402457, and 32582657 appeared [2] , the search for primitive trinomials corresponding to huge Mersenne primes continued. For degree 43112609, we have found four primitive trinomials (and their reciprocal):

^{43112609}+ x

^{3569337}+ 1, x

^{43112609}+ x

^{4463337}+ 1, x

^{43112609}+ x

^{17212521}+ 1, x

^{43112609}+ x

^{21078848}+ 1,

and for degree r = 42643801 , we have found exactly five:

^{r}+ x

^{55981}+ 1, x

^{r}+ x

^{3706066}+ 1, x

^{r}+ x

^{3896488}+ 1, x

^{r}+ x

^{12899278}+ 1, x

^{r}+ x

^{20150445}+ 1.

All those primitive trinomials have been checked by Allan Steel using Magma. Those results will be published in an invited paper to the AMS Notices.

Together with Will Orrick (Indiana University) and Judy-anne Osborn (Australian National University), Richard Brent and P. Zimmermann started another project on maximal determinants of Hadamard matrices. Results so far include the fact that the conjectured maximal determinant for n = 19 is the true maximal determinant, and similarly for n = 37 . This work was done with the support of the ANC associate team too.

Together with Philippe Dumas, Claude Gomez and Bruno Salvy, P. Zimmermann published an electronic version of the book “Calcul formel : mode d'emploi. Exemples en Maple” (in french), previously published by a commercial editor, and whose rights have been given back to the authors [12] .