763:
In 2015, for the first time, a periodic representation for any cubic irrational has been provided by means of ternary continued fractions, i.e., the problem of writing cubic irrationals as a periodic sequence of rational or integer numbers has been solved. However, the periodic representation does
592:
249:
128:
614:
proved that irrational numbers require an infinite sequence to express them as continued fractions. Moreover, this sequence is eventually periodic (again, so that there are natural numbers
689:) with the property that each sequence gives a unique real number and such that this real number belongs to the corresponding set if and only if the sequence is eventually periodic.
841:
algorithm) works for the totally real case only. The input for the algorithm is a triples of cubic vectors. A cubic vector is any vector generating a degree 3 extension of
861:
839:
891:
398:
441:
1150:
772:
183:
765:
1145:
814:
Two subtractive algorithms for finding a periodic representative of cubic vectors were proposed by Oleg
Karpenkov. The first (
729:
1140:
696:
asking if this situation could be generalised, that is can one assign a sequence of natural numbers to each real number
740:) that acted as a higher-dimensional analogue of continued fractions. He hoped to show that the sequence attached to (
676:
of degree 1, while quadratic irrationals are algebraic numbers that satisfy a polynomial of degree 2. For both these
145:
61:
764:
not derive from an algorithm defined over all real numbers and it is derived only starting from the knowledge of the
863:. In this case the cubic vectors are conjugate if and only if the output of the algorithm is periodic. The second (
944:
Extraits de lettres de M. Ch. Hermite à M. Jacobi sur différents objects de la théorie des nombres. (Continuation).
775:. This function ? : → also picks out quadratic irrational numbers since ?(
732:. Jacobi himself came up with an early example, finding a sequence corresponding to each pair of real numbers (
693:
704:
is a cubic irrational, that is an algebraic number of degree 3? Or, more generally, for each natural number
976:
General theory of continued-fraction-like algorithms in which each number is formed from three previous ones
52:
867:
algorithm) is conjectured to work for all cases (including for complex cubic vectors) and all dimensions
1081:
Karpenkov, Oleg (2022), "On
Hermite's problem, Jacobi–Perron type algorithms, and Dirichlet groups",
657:
844:
1117:
1090:
1046:
304:
817:
771:
Rather than generalising continued fractions, another approach to the problem is to generalise
1026:
677:
40:
39:, such that the sequence is eventually periodic precisely when the original number is a cubic
870:
1100:
1056:
951:
669:
587:{\displaystyle x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+\ddots }}}}}}.}
1112:
1068:
803:) is a non-dyadic rational number. Various generalisations of this function to either the
313:
262:
only if its decimal expansion is eventually periodic, that is if there are natural numbers
1108:
1064:
792:
259:
24:
911:
36:
1134:
993:
On the periodic writing of cubic irrationals and a generalization of RĂ©dei functions
1022:
968:
Allgemeine
Theorie der kettenbruchänlichen Algorithmen, in welche jede Zahl aus
804:
757:
173:, ... are integers between 0 and 9. Given this representation the number
28:
20:
1060:
708:
is there a way of assigning a sequence of natural numbers to each real number
673:
955:
1104:
760:, but was unable to do so and whether this is the case remains unsolved.
32:
808:
141:
1116:; see also Karpenkov's "On a periodic Jacobi-Perron type algorithm",
1051:
680:
of numbers we have a way to construct a sequence of natural numbers (
1121:
1095:
916:, Lausanne: Marcum-Michaelem Bousquet – via The Euler Archive
728:
Sequences that attempt to solve
Hermite's problem are often called
783:
is either rational or a quadratic irrational number, and moreover
611:
431:... are natural numbers. From this representation we can recover
1010:
Un
Algorithme pour l'approximation simultanée de Deux Granduers
999:(2015), no. 3, pp. 779-799, doi: 10.1142/S1793042115500438
244:{\displaystyle x=\sum _{n=0}^{\infty }{\frac {a_{n}}{10^{n}}}.}
811:
have been made, though none has yet solved
Hermite's problem.
700:
such that the sequence is eventually periodic precisely when
610:) terminates after finitely many terms. On the other hand,
1124:, merged into the published journal version of this paper.
545:
533:
512:
500:
479:
467:
548:
536:
515:
503:
482:
470:
303:
Another way of expressing numbers is to write them as
873:
847:
820:
444:
316:
186:
64:
1012:, Inaugural-Dissertation, Universität Zürich, 1905.
978:), Journal fĂĽr die reine und angewandte Mathematik
51:A standard way of writing real numbers is by their
946:, Journal fĂĽr die reine und angewandte Mathematik
885:
855:
833:
586:
392:
243:
122:
799:is a quadratic irrational precisely when ?(
123:{\displaystyle x=a_{0}.a_{1}a_{2}a_{3}\ldots \ }
748:) was eventually periodic if and only if both
8:
913:Introductio in analysin infinitorum, Vol. I
692:In 1848, Charles Hermite wrote a letter to
27:in 1848. He asked for a way of expressing
807: × or the two-dimensional
1094:
1050:
872:
849:
848:
846:
825:
819:
554:
549:
537:
530:
521:
516:
504:
497:
488:
483:
471:
464:
455:
443:
369:
356:
343:
330:
315:
230:
220:
214:
208:
197:
185:
108:
98:
88:
75:
63:
601:is a rational number then the sequence (
928:L'Ĺ“uvre scientifique de Charles Hermite
902:
1029:(2004), "A two-dimensional Minkowski
7:
730:multidimensional continued fractions
1151:Unsolved problems in number theory
787:is rational if and only if ?(
773:Minkowski's question-mark function
209:
14:
381:
323:
1:
930:, Ann. Sci. École Norm. Sup.
779:) is rational if and only if
972:vorhergehenden gebildet wird
856:{\displaystyle \mathbb {Q} }
1167:
950:(1850), pp.279–315,
1061:10.1016/j.jnt.2004.01.008
934:18 (1901), pp.9–34.
834:{\displaystyle \sin ^{2}}
768:of the cubic irrational.
1039:Journal of Number Theory
995:, Int. J. Number Theory
956:10.1515/crll.1850.40.279
910:Euler, Leonhard (1748),
694:Carl Gustav Jacob Jacobi
663:
1146:Algebraic number theory
982:(1868), pp.29–64.
886:{\displaystyle d\geq 3}
716:is algebraic of degree
712:that can pick out when
887:
857:
835:
588:
394:
245:
213:
124:
53:decimal representation
19:is an open problem in
888:
858:
836:
668:Rational numbers are
589:
395:
393:{\displaystyle x=,\ }
246:
193:
125:
1105:10.4064/aa210614-5-1
871:
845:
818:
658:quadratic irrational
622:such that for every
442:
314:
278:it is the case that
270:such that for every
184:
62:
1141:Continued fractions
626: ≥
547:
535:
514:
502:
481:
469:
305:continued fractions
274: ≥
883:
853:
831:
766:minimal polynomial
758:cubic number field
664:Hermite's question
652:), if and only if
584:
577:
572:
567:
542:
509:
476:
410:is an integer and
390:
241:
120:
966:C. G. J. Jacobi,
670:algebraic numbers
579:
574:
569:
546:
534:
513:
501:
480:
468:
389:
236:
119:
17:Hermite's problem
1158:
1125:
1115:
1098:
1083:Acta Arithmetica
1078:
1072:
1071:
1054:
1036:
1019:
1013:
1006:
1000:
989:
983:
964:
958:
941:
935:
924:
918:
917:
907:
892:
890:
889:
884:
862:
860:
859:
854:
852:
840:
838:
837:
832:
830:
829:
593:
591:
590:
585:
580:
578:
576:
575:
573:
571:
570:
568:
566:
559:
558:
543:
541:
531:
526:
525:
510:
508:
498:
493:
492:
477:
475:
465:
460:
459:
399:
397:
396:
391:
387:
374:
373:
361:
360:
348:
347:
335:
334:
254:The real number
250:
248:
247:
242:
237:
235:
234:
225:
224:
215:
212:
207:
129:
127:
126:
121:
117:
113:
112:
103:
102:
93:
92:
80:
79:
1166:
1165:
1161:
1160:
1159:
1157:
1156:
1155:
1131:
1130:
1129:
1128:
1080:
1079:
1075:
1030:
1027:Garrity, Thomas
1023:Beaver, Olga R.
1021:
1020:
1016:
1007:
1003:
990:
986:
965:
961:
942:
938:
925:
921:
909:
908:
904:
899:
869:
868:
843:
842:
821:
816:
815:
793:dyadic rational
726:
688:
672:that satisfy a
666:
651:
642:
609:
550:
544:
532:
517:
511:
499:
484:
478:
466:
451:
440:
439:
430:
423:
416:
409:
365:
352:
339:
326:
312:
311:
299:
290:
260:rational number
226:
216:
182:
181:
172:
165:
158:
139:
104:
94:
84:
71:
60:
59:
49:
37:natural numbers
25:Charles Hermite
12:
11:
5:
1164:
1162:
1154:
1153:
1148:
1143:
1133:
1132:
1127:
1126:
1073:
1045:(1): 105–134,
1014:
1001:
984:
959:
936:
926:Émile Picard,
919:
901:
900:
898:
895:
882:
879:
876:
851:
828:
824:
756:belonged to a
725:
722:
684:
665:
662:
647:
634:
605:
595:
594:
583:
565:
562:
557:
553:
540:
529:
524:
520:
507:
496:
491:
487:
474:
463:
458:
454:
450:
447:
428:
421:
414:
407:
401:
400:
386:
383:
380:
377:
372:
368:
364:
359:
355:
351:
346:
342:
338:
333:
329:
325:
322:
319:
295:
282:
252:
251:
240:
233:
229:
223:
219:
211:
206:
203:
200:
196:
192:
189:
170:
163:
156:
137:
131:
130:
116:
111:
107:
101:
97:
91:
87:
83:
78:
74:
70:
67:
48:
45:
13:
10:
9:
6:
4:
3:
2:
1163:
1152:
1149:
1147:
1144:
1142:
1139:
1138:
1136:
1123:
1119:
1114:
1110:
1106:
1102:
1097:
1092:
1088:
1084:
1077:
1074:
1070:
1066:
1062:
1058:
1053:
1048:
1044:
1040:
1034:
1028:
1024:
1018:
1015:
1011:
1005:
1002:
998:
994:
991:Nadir Murru,
988:
985:
981:
977:
973:
969:
963:
960:
957:
953:
949:
945:
940:
937:
933:
929:
923:
920:
915:
914:
906:
903:
896:
894:
880:
877:
874:
866:
826:
822:
812:
810:
806:
802:
798:
794:
790:
786:
782:
778:
774:
769:
767:
761:
759:
755:
751:
747:
743:
739:
735:
731:
723:
721:
719:
715:
711:
707:
703:
699:
695:
690:
687:
683:
679:
675:
671:
661:
659:
655:
650:
646:
643: =
641:
637:
633:
629:
625:
621:
617:
613:
608:
604:
600:
581:
563:
560:
555:
551:
538:
527:
522:
518:
505:
494:
489:
485:
472:
461:
456:
452:
448:
445:
438:
437:
436:
434:
427:
420:
413:
406:
384:
378:
375:
370:
366:
362:
357:
353:
349:
344:
340:
336:
331:
327:
320:
317:
310:
309:
308:
306:
301:
298:
294:
291: =
289:
285:
281:
277:
273:
269:
265:
261:
257:
238:
231:
227:
221:
217:
204:
201:
198:
194:
190:
187:
180:
179:
178:
176:
169:
162:
155:
151:
147:
143:
136:
114:
109:
105:
99:
95:
89:
85:
81:
76:
72:
68:
65:
58:
57:
56:
54:
46:
44:
42:
38:
34:
30:
26:
22:
18:
1089:(1): 27–48,
1086:
1082:
1076:
1052:math/0210480
1042:
1038:
1032:
1017:
1009:
1008:L. Kollros,
1004:
996:
992:
987:
979:
975:
971:
967:
962:
947:
943:
939:
931:
927:
922:
912:
905:
864:
813:
800:
796:
788:
784:
780:
776:
770:
762:
753:
749:
745:
741:
737:
733:
727:
717:
713:
709:
705:
701:
697:
691:
685:
681:
667:
653:
648:
644:
639:
635:
631:
627:
623:
619:
615:
606:
602:
598:
596:
432:
425:
418:
411:
404:
402:
302:
296:
292:
287:
283:
279:
275:
271:
267:
263:
255:
253:
177:is equal to
174:
167:
160:
153:
149:
146:integer part
134:
132:
50:
29:real numbers
16:
15:
1037:function",
805:unit square
55:, such as:
21:mathematics
1135:Categories
1122:2101.12627
1096:2101.12707
974:(English:
897:References
724:Approaches
674:polynomial
47:Motivation
41:irrational
878:≥
564:⋱
379:…
307:, as in:
210:∞
195:∑
115:…
33:sequences
23:posed by
744:,
736:,
630:we have
1113:4415995
1069:2059953
809:simplex
795:, thus
791:) is a
142:integer
1111:
1067:
435:since
403:where
388:
152:, and
144:, the
140:is an
133:where
118:
1118:arXiv
1091:arXiv
1047:arXiv
656:is a
612:Euler
258:is a
970:drei
865:HAPD
752:and
678:sets
618:and
266:and
1101:doi
1087:203
1057:doi
1043:107
952:doi
823:sin
597:If
148:of
35:of
31:as
1137::
1109:MR
1107:,
1099:,
1085:,
1065:MR
1063:,
1055:,
1041:,
1031:?(
1025:;
997:11
980:69
948:40
893:.
720:?
660:.
424:,
417:,
300:.
228:10
166:,
159:,
43:.
1120::
1103::
1093::
1059::
1049::
1035:)
1033:x
954::
932:3
881:3
875:d
850:Q
827:2
801:x
797:x
789:x
785:x
781:x
777:x
754:y
750:x
746:y
742:x
738:y
734:x
718:d
714:x
710:x
706:d
702:x
698:x
686:n
682:a
654:x
649:n
645:a
640:p
638:+
636:n
632:a
628:N
624:n
620:p
616:N
607:n
603:a
599:x
582:.
561:+
556:3
552:a
539:1
528:+
523:2
519:a
506:1
495:+
490:1
486:a
473:1
462:+
457:0
453:a
449:=
446:x
433:x
429:3
426:a
422:2
419:a
415:1
412:a
408:0
405:a
385:,
382:]
376:,
371:3
367:a
363:,
358:2
354:a
350:,
345:1
341:a
337:;
332:0
328:a
324:[
321:=
318:x
297:n
293:a
288:p
286:+
284:n
280:a
276:N
272:n
268:p
264:N
256:x
239:.
232:n
222:n
218:a
205:0
202:=
199:n
191:=
188:x
175:x
171:3
168:a
164:2
161:a
157:1
154:a
150:x
138:0
135:a
110:3
106:a
100:2
96:a
90:1
86:a
82:.
77:0
73:a
69:=
66:x
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