353:
911:
813:
252:
621:
714:
109:
555:
503:
678:
819:
153:
are allowed (here complex numbers are not regarded as rational when they have an imaginary part not equal to 0, even if both the real and imaginary parts are rational).
723:
1111:
1078:
1009:
179:. (This is the multivalued inverse of the exponential function exp.) This accounts for the phrase "any value of" in the theorem's statement.
224:
over the rationals, then they are linearly independent over the algebraic numbers. The generalisation of this statement to more general
1192:
934:
1137:
1043:
348:{\displaystyle {\left({\sqrt {2}}^{\sqrt {2}}\right)}^{\sqrt {2}}={\sqrt {2}}^{{\sqrt {2}}\cdot {\sqrt {2}}}={\sqrt {2}}^{2}=2.}
1187:
649:
1027:
922:
229:
630:
1001:
993:
225:
560:
945:
683:
1172:
436:
124:
30:
718:
79:
906:{\displaystyle i^{i}=\left(e^{\frac {i\pi }{2}}\right)^{i}=e^{-{\frac {\pi }{2}}}=0.207879576\ldots }
508:
172:
459:
948:; if proven it would imply both the Gelfond–Schneider theorem and the Lindemann–Weierstrass theorem
654:
221:
1095:
1070:
370:, which (as proven by the theorem itself) is transcendental rather than algebraic. Similarly, if
939:
1148:
1133:
1107:
1074:
1060:
1039:
1005:
440:
176:
46:
42:
1015:
70:
1088:
1053:
1084:
1049:
1035:
1019:
116:
1100:
150:
67:
972:
Bulletin de l'Académie des
Sciences de l'URSS. Classe des sciences mathématiques et na
1181:
1151:
1126:
1121:
412:
243:
be algebraic is removed, the statement does not remain true in general. For example,
1064:
808:{\displaystyle e^{\pi }=\left(e^{i\pi }\right)^{-i}=(-1)^{-i}=23.14069263\ldots }
645:
The transcendence of the following numbers follows immediately from the theorem:
408:
146:
22:
206:
is either rational or transcendental. This may be expressed as saying that if
1156:
190:
are nonzero algebraic numbers, and we take any non-zero logarithm of
967:
182:
An equivalent formulation of the theorem is the following: if
1034:, Encyclopedia of mathematical sciences, vol. 44,
822:
726:
686:
657:
563:
511:
462:
255:
82:
921:
The
Gelfond–Schneider theorem answers affirmatively
391:is algebraic. A characterization of the values for
1125:
1099:
905:
807:
708:
672:
615:
549:
497:
347:
103:
41:It was originally proved independently in 1934 by
228:of several algebraic numbers is in the domain of
18:On the transcendence of a large class of numbers
623:is either rational or transcendental, where log
8:
98:
86:
1102:Topics in Number Theory, Volumes I and II
882:
878:
865:
845:
827:
821:
787:
762:
749:
731:
725:
695:
688:
685:
662:
656:
616:{\displaystyle (\log _{p}a)/(\log _{p}b)}
598:
586:
571:
562:
532:
527:
512:
510:
483:
478:
463:
461:
333:
326:
313:
303:
302:
295:
283:
270:
263:
257:
254:
81:
1173:A proof of the Gelfond–Schneider theorem
709:{\displaystyle {\sqrt {2}}^{\sqrt {2}}.}
1132:. Mathematical Association of America.
958:
968:"Sur le septième Problème de Hilbert"
7:
1069:, Dover Phoenix editions, New York:
1066:Transcendental and algebraic numbers
175:, where log stands for the complex
14:
1106:. New York: Dover Publications.
384:, which is transcendental, then
104:{\displaystyle \not \in \{0,1\}}
550:{\displaystyle |b-1|_{p}<1,}
784:
774:
610:
591:
583:
564:
528:
513:
498:{\displaystyle |a-1|_{p}<1}
479:
464:
452:, and they are algebraic over
1:
935:Lindemann–Weierstrass theorem
673:{\displaystyle 2^{\sqrt {2}}}
399:which yield a transcendental
33:of a large class of numbers.
998:Transcendental number theory
942:; an extension of the result
418:analogue of the theorem: if
230:transcendental number theory
1152:"Gelfond-Schneider Theorem"
1211:
1002:Cambridge University Press
966:Aleksandr Gelfond (1934).
650:Gelfond–Schneider constant
226:linear forms in logarithms
1193:Theorems in number theory
923:Hilbert's seventh problem
27:Gelfond–Schneider theorem
634:-adic logarithm function
235:If the restriction that
1188:Transcendental numbers
1032:Transcendental numbers
907:
809:
710:
674:
617:
551:
499:
349:
145:are not restricted to
105:
946:Schanuel's conjecture
908:
810:
711:
675:
618:
552:
500:
350:
125:transcendental number
106:
820:
724:
684:
680:and its square root
655:
561:
509:
460:
253:
222:linearly independent
119:, then any value of
80:
1096:LeVeque, William J.
1149:Weisstein, Eric W.
1128:Irrational Numbers
1071:Dover Publications
1028:Nesterenko, Yu. V.
903:
805:
719:Gelfond's constant
706:
670:
613:
547:
495:
345:
101:
1113:978-0-486-42539-9
1080:978-0-486-49526-2
1011:978-0-521-20461-3
890:
858:
700:
693:
667:
441:algebraic closure
382:= (log 2)/(log 3)
331:
318:
308:
300:
288:
275:
268:
177:natural logarithm
71:algebraic numbers
47:Theodor Schneider
43:Aleksandr Gelfond
1200:
1162:
1161:
1143:
1131:
1117:
1105:
1091:
1056:
1026:Feldman, N. I.;
1022:
980:
979:
963:
912:
910:
909:
904:
893:
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870:
869:
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832:
831:
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811:
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795:
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770:
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736:
735:
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696:
694:
689:
679:
677:
676:
671:
669:
668:
663:
622:
620:
619:
614:
603:
602:
590:
576:
575:
556:
554:
553:
548:
537:
536:
531:
516:
504:
502:
501:
496:
488:
487:
482:
467:
390:
383:
376:
369:
368:
354:
352:
351:
346:
338:
337:
332:
327:
321:
320:
319:
314:
309:
304:
301:
296:
290:
289:
284:
282:
281:
277:
276:
271:
269:
264:
219:
212:
205:
170:
110:
108:
107:
102:
29:establishes the
1210:
1209:
1203:
1202:
1201:
1199:
1198:
1197:
1178:
1177:
1169:
1147:
1146:
1140:
1120:
1114:
1094:
1081:
1061:Gel'fond, A. O.
1059:
1046:
1036:Springer-Verlag
1025:
1012:
992:
989:
987:Further reading
984:
983:
965:
964:
960:
955:
940:Baker's theorem
931:
919:
874:
847:
841:
837:
836:
823:
818:
817:
783:
745:
741:
740:
727:
722:
721:
687:
682:
681:
658:
653:
652:
643:
628:
594:
567:
559:
558:
526:
507:
506:
477:
458:
457:
451:
434:
385:
378:
371:
366:
364:
325:
294:
262:
258:
256:
251:
250:
214:
207:
195:
157:
151:complex numbers
134:
78:
77:
55:
39:
19:
12:
11:
5:
1208:
1207:
1204:
1196:
1195:
1190:
1180:
1179:
1176:
1175:
1168:
1167:External links
1165:
1164:
1163:
1144:
1138:
1118:
1112:
1092:
1079:
1057:
1044:
1023:
1010:
1004:, p. 10,
988:
985:
982:
981:
957:
956:
954:
951:
950:
949:
943:
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930:
927:
918:
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835:
830:
826:
815:
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404:
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317:
312:
307:
299:
293:
287:
280:
274:
267:
261:
245:
244:
233:
180:
154:
137:The values of
133:
130:
129:
128:
100:
97:
94:
91:
88:
85:
54:
51:
38:
35:
17:
13:
10:
9:
6:
4:
3:
2:
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1205:
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1191:
1189:
1186:
1185:
1183:
1174:
1171:
1170:
1166:
1159:
1158:
1153:
1150:
1145:
1141:
1139:0-88385-011-7
1135:
1130:
1129:
1123:
1119:
1115:
1109:
1104:
1103:
1097:
1093:
1090:
1086:
1082:
1076:
1072:
1068:
1067:
1062:
1058:
1055:
1051:
1047:
1045:3-540-61467-2
1041:
1037:
1033:
1029:
1024:
1021:
1017:
1013:
1007:
1003:
999:
995:
991:
990:
986:
978:(4): 623–634.
977:
973:
969:
962:
959:
952:
947:
944:
941:
938:
936:
933:
932:
928:
926:
924:
916:
900:
897:
894:
887:
884:
879:
875:
871:
866:
861:
855:
851:
848:
842:
838:
833:
828:
824:
816:
802:
799:
796:
791:
788:
780:
777:
771:
766:
763:
758:
753:
750:
746:
742:
737:
732:
728:
720:
717:
703:
697:
690:
664:
659:
651:
648:
647:
646:
640:
635:
633:
627:
607:
604:
599:
595:
587:
580:
577:
572:
568:
544:
541:
538:
533:
523:
520:
517:
492:
489:
484:
474:
471:
468:
455:
450:
446:
442:
438:
433:
429:
425:
421:
417:
415:
410:
407:
406:
403:is not known.
402:
398:
394:
388:
381:
374:
362:
358:
342:
339:
334:
328:
322:
315:
310:
305:
297:
291:
285:
278:
272:
265:
259:
249:
248:
247:
246:
242:
238:
234:
231:
227:
223:
218:
211:
203:
199:
193:
189:
185:
181:
178:
174:
168:
164:
160:
155:
152:
148:
144:
140:
136:
135:
131:
126:
122:
118:
114:
95:
92:
89:
83:
76:
72:
69:
65:
61:
57:
56:
52:
50:
48:
44:
36:
34:
32:
31:transcendence
28:
24:
16:
1155:
1127:
1101:
1065:
1031:
997:
975:
971:
961:
920:
917:Applications
644:
631:
625:
453:
448:
444:
431:
427:
423:
419:
413:
400:
396:
392:
386:
379:
372:
360:
240:
236:
216:
209:
201:
197:
191:
187:
183:
166:
162:
158:
156:In general,
147:real numbers
142:
138:
120:
112:
74:
63:
59:
40:
26:
20:
15:
1122:Niven, Ivan
994:Baker, Alan
898:0.207879576
800:23.14069263
641:Corollaries
411:proved the
409:Kurt Mahler
173:multivalued
23:mathematics
1182:Categories
1020:0297.10013
953:References
437:completion
1157:MathWorld
1098:(2002) .
1063:(1960) ,
901:…
885:π
880:−
852:π
803:…
789:−
778:−
764:−
754:π
733:π
605:
578:
521:−
472:−
456:, and if
311:⋅
53:Statement
1124:(1956).
1030:(1998),
996:(1975),
929:See also
132:Comments
117:rational
84:∉
1089:0057921
1054:1603604
629:is the
439:of the
426:are in
365:√
200:)/(log
194:, then
68:complex
37:History
1136:
1110:
1087:
1077:
1052:
1042:
1018:
1008:
435:, the
359:Here,
161:= exp(
25:, the
557:then
416:-adic
196:(log
123:is a
73:with
1134:ISBN
1108:ISBN
1075:ISBN
1040:ISBN
1006:ISBN
539:<
505:and
490:<
422:and
395:and
377:and
239:and
220:are
215:log
208:log
186:and
165:log
141:and
115:not
111:and
66:are
62:and
45:and
1016:Zbl
976:VII
596:log
569:log
443:of
389:= 2
375:= 3
363:is
171:is
58:If
49:.
21:In
1184::
1154:.
1085:MR
1083:,
1073:,
1050:MR
1048:,
1038:,
1014:,
1000:,
974:.
970:.
925:.
343:2.
213:,
149:;
1160:.
1142:.
1116:.
895:=
888:2
876:e
872:=
867:i
862:)
856:2
849:i
843:e
839:(
834:=
829:i
825:i
797:=
792:i
785:)
781:1
775:(
772:=
767:i
759:)
751:i
747:e
743:(
738:=
729:e
704:.
698:2
691:2
665:2
660:2
636:.
632:p
626:p
611:)
608:b
600:p
592:(
588:/
584:)
581:a
573:p
565:(
545:,
542:1
534:p
529:|
524:1
518:b
514:|
493:1
485:p
480:|
475:1
469:a
465:|
454:Q
449:p
445:Q
432:p
428:C
424:b
420:a
414:p
401:a
397:b
393:a
387:a
380:b
373:a
367:2
361:a
340:=
335:2
329:2
323:=
316:2
306:2
298:2
292:=
286:2
279:)
273:2
266:2
260:(
241:b
237:a
232:.
217:γ
210:α
204:)
202:α
198:γ
192:α
188:γ
184:α
169:)
167:a
163:b
159:a
143:b
139:a
127:.
121:a
113:b
99:}
96:1
93:,
90:0
87:{
75:a
64:b
60:a
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