947:
927:
917:
494:
431:
942:
937:
324:
300:
244:
211:
526:
503:
is again the exponent of a
Mersenne prime. The highest degree trinomials found were three trinomials of degree 74,207,281, also a Mersenne prime exponent.
907:
580:
922:
698:
518:
377:
912:
605:
818:
838:
814:
373:
185:
147:
139:
99:
629:(1975). Traub, J. F. (ed.). "Multiple-Precision Zero-Finding Methods and the Complexity of Elementary Function Evaluation".
280:
887:
538:
530:
366:
354:
189:
511:
453:
247:
390:
159:
447:
In 2009 and 2016, Brent and Paul
Zimmermann discovered some even larger primitive trinomials, for example:
932:
634:
437:
174:
167:
577:
600:
Prentice-Hall, Englewood Cliffs, NJ. Reprinted by Dover
Publications, Mineola, New York, 2002 and 2013.
265:
163:
143:
558:
902:
778:
762:
718:
654:
626:
546:
251:
639:
514:), a book about algorithms for performing arithmetic, and their implementation on modern computers.
500:
215:
55:
796:
744:
680:
542:
269:
261:
135:
601:
178:
309:
285:
786:
734:
670:
106:
89:
584:
303:
229:
196:
115:
17:
782:
842:
609:
441:
276:
258:
896:
358:
155:
151:
131:
111:
534:
381:
365:
algorithm. He later factored the tenth and eleventh Fermat numbers using
Lenstra's
65:
791:
766:
362:
85:
883:
45:
878:
800:
748:
684:
343:
268:
lie on the critical line, providing some experimental evidence for the
177:(an algorithm for solving equations numerically) which is now known as
699:
Some New
Algorithms for High-Precision Computation of Euler's Constant
739:
722:
675:
658:
862:
659:"On the Zeros of the Riemann Zeta Function in the Critical Strip"
522:
226:) etc.) can be evaluated to high precision in the same time as
613:
948:
Fellows of the
Society for Industrial and Applied Mathematics
279:
found a new algorithm for high-precision computation of the
193:
928:
1995 fellows of the
Association for Computing Machinery
456:
393:
312:
288:
232:
199:
918:
Academic staff of the
Australian National University
697:
Brent, Richard Peirce and McMillan, E. M. (1980). "
105:
95:
81:
61:
51:
41:
34:
488:
425:
318:
294:
238:
205:
612:is available on his own professional web page at
598:Algorithms for Minimization without Derivatives.
246:(apart from a small constant factor) using the
27:Australian mathematician and computer scientist
506:In 2011, Brent and Paul Zimmermann published
8:
943:Fellows of the Australian Academy of Science
938:People from the Australian Capital Territory
847:Notices of the American Mathematical Society
578:Federation Fellowships Funding Outcomes 2004
257:In 1979 he showed that the first 75 million
723:"Factorization of the Eighth Fermat Number"
489:{\displaystyle x^{43112609}+x^{3569337}+1.}
214:. At the same time, he showed that all the
813:Brent, Richard Peirce and Larvala, S. and
767:"Factorization of the Tenth Fermat Number"
426:{\displaystyle x^{6972593}+x^{3037958}+1.}
142:. From March 2005 to March 2010 he was a
31:
790:
738:
674:
638:
474:
461:
455:
411:
398:
392:
311:
287:
231:
198:
863:"Twelve new primitive binary trinomials"
192:, used in high-precision calculation of
819:A primitive trinomial of degree 6972593
570:
350:is extremely large (greater than 10).
633:. New York: Academic Press: 151–176.
138:. He is an emeritus professor at the
7:
326:can not have a simple rational form
861:Richard P. Brent, Paul Zimmermann,
519:Association for Computing Machinery
372:In 2002, Brent, Samuli Larvala and
25:
631:Analytic Computational Complexity
150:. His research interests include
865:, arXiv:1605.09213, 24 May 2016.
908:Australian computer scientists
541:. In 2014, he was awarded the
533:. In 2005, he was awarded the
275:In 1980 he and Nobel laureate
148:Australian National University
140:Australian National University
100:Australian National University
1:
888:Mathematics Genealogy Project
792:10.1090/s0025-5718-99-00992-8
596:Richard Peirce Brent (1973).
587:. Australian Research Council
539:Australian Academy of Science
531:Australian Academy of Science
440:6972593 is the exponent of a
367:elliptic curve factorisation
188:independently conceived the
964:
923:Complex systems scientists
837:Brent, Richard Peirce and
823:Mathematics of Computation
771:Mathematics of Computation
727:Mathematics of Computation
703:Mathematics of Computation
663:Mathematics of Computation
512:Cambridge University Press
508:Modern Computer Arithmetic
913:Australian mathematicians
879:Richard Brent's home page
721:; Pollard, J. M. (1981).
517:Brent is a Fellow of the
281:Euler–Mascheroni constant
248:arithmetic-geometric mean
121:
74:
18:Richard Brent (scientist)
843:The great trinomial hunt
376:discovered a very large
173:In 1973, he published a
160:random number generators
361:using a variant of the
319:{\displaystyle \gamma }
295:{\displaystyle \gamma }
190:Salamin–Brent algorithm
490:
427:
320:
296:
240:
207:
175:root-finding algorithm
168:analysis of algorithms
763:Brent, Richard Peirce
719:Brent, Richard Peirce
655:Brent, Richard Peirce
627:Brent, Richard Peirce
491:
428:
321:
297:
266:Riemann zeta function
241:
208:
164:computer architecture
547:Macquarie University
454:
391:
357:factored the eighth
310:
286:
252:Carl Friedrich Gauss
239:{\displaystyle \pi }
230:
216:elementary functions
206:{\displaystyle \pi }
197:
128:Richard Peirce Brent
36:Richard Peirce Brent
783:1999MaCom..68..429B
378:primitive trinomial
56:Stanford University
669:(148): 1361–1372.
583:2012-07-07 at the
486:
423:
316:
306:, and showed that
292:
270:Riemann hypothesis
236:
203:
136:computer scientist
144:Federation Fellow
130:is an Australian
125:
124:
107:Doctoral advisors
76:Scientific career
70:
16:(Redirected from
955:
884:Richard P. Brent
866:
859:
853:
839:Zimmermann, Paul
835:
829:
828:(250) 1001-1002.
815:Zimmermann, Paul
811:
805:
804:
794:
777:(225): 429–451.
759:
753:
752:
742:
733:(154): 627–630.
715:
709:
695:
689:
688:
678:
651:
645:
644:
642:
623:
617:
610:Original edition
594:
588:
575:
559:Brent–Kung adder
495:
493:
492:
487:
479:
478:
466:
465:
432:
430:
429:
424:
416:
415:
403:
402:
325:
323:
322:
317:
304:Bessel functions
301:
299:
298:
293:
245:
243:
242:
237:
212:
210:
209:
204:
90:computer science
68:
32:
21:
963:
962:
958:
957:
956:
954:
953:
952:
893:
892:
875:
870:
869:
860:
856:
836:
832:
812:
808:
761:
760:
756:
740:10.2307/2007666
717:
716:
712:
696:
692:
676:10.2307/2006473
653:
652:
648:
640:10.1.1.119.3317
625:
624:
620:
595:
591:
585:Wayback Machine
576:
572:
567:
555:
470:
457:
452:
451:
407:
394:
389:
388:
374:Paul Zimmermann
353:In 1980 he and
308:
307:
284:
283:
228:
227:
195:
194:
184:In 1975 he and
154:(in particular
116:George Forsythe
114:
52:Alma mater
37:
28:
23:
22:
15:
12:
11:
5:
961:
959:
951:
950:
945:
940:
935:
930:
925:
920:
915:
910:
905:
895:
894:
891:
890:
881:
874:
873:External links
871:
868:
867:
854:
830:
806:
754:
710:
708:(149) 305-312.
690:
646:
618:
589:
569:
568:
566:
563:
562:
561:
554:
551:
497:
496:
485:
482:
477:
473:
469:
464:
460:
442:Mersenne prime
434:
433:
422:
419:
414:
410:
406:
401:
397:
315:
291:
277:Edwin McMillan
235:
202:
186:Eugene Salamin
179:Brent's method
123:
122:
119:
118:
109:
103:
102:
97:
93:
92:
83:
79:
78:
72:
71:
63:
59:
58:
53:
49:
48:
43:
39:
38:
35:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
960:
949:
946:
944:
941:
939:
936:
934:
933:Living people
931:
929:
926:
924:
921:
919:
916:
914:
911:
909:
906:
904:
901:
900:
898:
889:
885:
882:
880:
877:
876:
872:
864:
858:
855:
851:
848:
844:
840:
834:
831:
827:
824:
820:
816:
810:
807:
802:
798:
793:
788:
784:
780:
776:
772:
768:
764:
758:
755:
750:
746:
741:
736:
732:
728:
724:
720:
714:
711:
707:
704:
700:
694:
691:
686:
682:
677:
672:
668:
664:
660:
656:
650:
647:
641:
636:
632:
628:
622:
619:
615:
611:
607:
606:0-486-41998-3
603:
599:
593:
590:
586:
582:
579:
574:
571:
564:
560:
557:
556:
552:
550:
548:
544:
540:
536:
532:
528:
524:
520:
515:
513:
509:
504:
502:
483:
480:
475:
471:
467:
462:
458:
450:
449:
448:
445:
443:
439:
420:
417:
412:
408:
404:
399:
395:
387:
386:
385:
383:
379:
375:
370:
368:
364:
360:
359:Fermat number
356:
351:
349:
345:
341:
337:
333:
329:
313:
305:
289:
282:
278:
273:
271:
267:
263:
260:
255:
253:
249:
233:
225:
221:
218:(such as log(
217:
213:
200:
191:
187:
182:
180:
176:
171:
169:
165:
161:
157:
156:factorisation
153:
152:number theory
149:
145:
141:
137:
133:
132:mathematician
129:
120:
117:
113:
112:Gene H. Golub
110:
108:
104:
101:
98:
94:
91:
87:
84:
80:
77:
73:
67:
64:
60:
57:
54:
50:
47:
44:
40:
33:
30:
19:
857:
849:
846:
833:
825:
822:
809:
774:
770:
757:
730:
726:
713:
705:
702:
693:
666:
662:
649:
630:
621:
597:
592:
573:
535:Hannan Medal
516:
507:
505:
498:
446:
435:
371:
355:John Pollard
352:
347:
339:
335:
331:
327:
274:
256:
223:
219:
183:
172:
127:
126:
96:Institutions
75:
66:Hannan Medal
29:
903:1946 births
543:Moyal Medal
499:The degree
369:algorithm.
363:Pollard rho
86:Mathematics
42:Nationality
897:Categories
565:References
46:Australian
841:(2011). "
817:(2005). "
635:CiteSeerX
346:) unless
314:γ
290:γ
234:π
201:π
852:233-239.
765:(1999).
657:(1979).
581:Archived
553:See also
529:and the
501:43112609
463:43112609
344:integers
886:at the
801:2585124
779:Bibcode
749:2007666
685:2006473
537:by the
476:3569337
413:3037958
400:6972593
334:(where
264:of the
259:complex
222:), sin(
146:at the
799:
747:
683:
637:
604:
521:, the
438:degree
302:using
166:, and
82:Fields
69:(2005)
62:Awards
797:JSTOR
745:JSTOR
681:JSTOR
384:(2):
380:over
262:zeros
602:ISBN
527:SIAM
523:IEEE
436:The
342:are
338:and
134:and
845:".
821:".
787:doi
735:doi
701:".
671:doi
614:ANU
545:by
250:of
158:),
899::
850:58
826:74
795:.
785:.
775:68
773:.
769:.
743:.
731:36
729:.
725:.
706:34
679:.
667:33
665:.
661:.
608:.
549:.
525:,
484:1.
444:.
421:1.
382:GF
272:.
254:.
181:.
170:.
162:,
88:,
803:.
789::
781::
751:.
737::
687:.
673::
643:.
616:.
510:(
481:+
472:x
468:+
459:x
418:+
409:x
405:+
396:x
348:q
340:q
336:p
332:q
330:/
328:p
224:x
220:x
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
