38:
177:
This version excludes 7 = 2+2+3 because this requires the even prime 2. On odd numbers larger than 7 it is slightly stronger as it also excludes sums like 17 = 2+2+13, which are allowed in the other formulation. Helfgott's proof covers both versions of the conjecture. Like the other formulation, this
317:
implies
Goldbach's weak conjecture for all numbers. This result combines a general statement valid for numbers greater than 10 with an extensive computer search of the small cases. Saouter also conducted a computer search covering the same cases at approximately the same time.
153:(concerning sums of two primes) is proven, then this would also be true. For if every even number greater than 4 is the sum of two odd primes, adding 3 to each even number greater than 4 will produce the odd numbers greater than 7 (and 7 itself is equal to 2+2+3).
696:
407:
is still much too large to admit checking all smaller numbers by computer. (Computer searches have only reached as far as 10 for the strong
Goldbach conjecture, and not much further than that for the weak Goldbach conjecture.)
547:
401:
300:
208:
The weak conjecture is simply this statement restricted to the case where the integer is odd (and possibly with the added requirement that the three primes in the sum be odd).
638:
441:
594:
1178:
614:
166:
series in 2015, and has been undergoing further review and revision since; fully-refereed chapters in close to final form are being made public in the process.
643:
302:
is large enough. The integer part of this number has 4,008,660 decimal digits, so checking every number under this figure would be completely infeasible.
1378:
1373:
1223:
1091:
1206:
1171:
1201:
1101:
1342:
416:
1258:
1383:
1164:
1125:
1196:
895:
1368:
1363:
446:
314:
225:
354:
200:. One formulation of the strong Goldbach conjecture, equivalent to the more common one in terms of sums of two primes, is
1228:
251:
1263:
245:
1337:
233:
162:
1213:
1322:
1268:
244:
odd numbers can be expressed as the sum of three primes. Vinogradov's original proof, as it used the ineffective
1248:
1327:
1295:
1283:
187:
146:
97:
1312:
1253:
1233:
1218:
348:
237:
221:
1307:
1238:
306:
1300:
1273:
1082:
1317:
1278:
619:
422:
333:
552:
1332:
755:
419:
estimates sufficiently to unconditionally prove the weak
Goldbach conjecture. Here, the major arcs
1243:
1140:
1119:
1061:
1040:
1021:
995:
830:
780:
735:
337:
241:
229:
193:
61:
160:
released a proof of
Goldbach's weak conjecture. The proof was accepted for publication in the
1107:
1097:
850:
Deshouillers, Jean-Marc; Effinger, Gove W.; Te Riele, Herman J. J.; Zinoviev, Dmitrii (1997).
805:
718:
321:
248:, did not give a bound for "sufficiently large"; his student K. Borozdkin (1956) derived that
1005:
957:
948:
911:
866:
1017:
971:
925:
880:
1013:
985:
Tao, Terence (2014). "Every odd number greater than 1 is the sum of at most five primes".
967:
921:
876:
412:
328:≥ 4 is in fact the sum of at most six primes, from which it follows that every odd number
310:
157:
79:
715:
Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIème siècle
236:
eliminated the dependency on the generalised
Riemann hypothesis and proved directly (see
808:
599:
197:
17:
940:
37:
1357:
106:
51:
1009:
691:{\displaystyle {\mathfrak {m}}=(\mathbb {R} /\mathbb {Z} )\setminus {\mathfrak {M}}}
1025:
134:
916:
871:
987:
903:
341:
217:
173:
Every odd number greater than 7 can be expressed as the sum of three odd primes.
142:
130:
1111:
962:
813:
1156:
404:
344:
proved this without the
Riemann Hypothesis; this improves both results.
851:
734:
Helfgott, Harald A. (2013). "The ternary
Goldbach conjecture is true".
204:
Every integer greater than 5 can be written as the sum of three primes.
859:
Electronic
Research Announcements of the American Mathematical Society
42:
Letter from
Goldbach to Euler dated on 7 June 1742 (Latin-German)
852:"A complete Vinogradov 3-primes theorem under the Riemann hypothesis"
336:
showed every odd integer is a sum of at most five primes, under the
1145:
835:
1066:
1045:
1000:
740:
351:) and Wang Tian-Ze lowered Borozdkin's threshold to approximately
1060:
Helfgott, Harald A. (2013). "Major arcs for
Goldbach's theorem".
1039:
Helfgott, Harald A. (2012). "Minor arcs for Goldbach's problem".
829:
Helfgott, Harald Andrés (2015). "The ternary Goldbach problem".
178:
one also immediately follows from Goldbach's strong conjecture.
1160:
1139:
Helfgott, Harald A. (2015). "The ternary Goldbach problem".
1096:. Vol. 2. Seoul, KOR: Kyung Moon SA. pp. 391–418.
542:{\displaystyle \left(a/q-cr_{0}/qx,a/q+cr_{0}/qx\right)}
137:. (A prime may be used more than once in the same sum.)
941:"On Ĺ nirelman's constant under the Riemann hypothesis"
396:{\displaystyle n>e^{3100}\approx 2\times 10^{1346}}
646:
622:
602:
555:
449:
425:
357:
254:
192:
The conjecture originated in correspondence between
133:
greater than 5 can be expressed as the sum of three
93:
85:
75:
67:
57:
47:
690:
632:
608:
588:
541:
435:
395:
294:
313:and Zinoviev published a result showing that the
295:{\displaystyle e^{e^{16.038}}\approx 3^{3^{15}}}
896:"Checking the odd Goldbach Conjecture up to 10"
228:, the weak Goldbach conjecture is true for all
1172:
8:
30:
1179:
1165:
1157:
36:
29:
1144:
1065:
1044:
999:
961:
915:
870:
834:
739:
682:
681:
671:
670:
665:
661:
660:
648:
647:
645:
624:
623:
621:
601:
580:
559:
554:
523:
517:
499:
482:
476:
458:
448:
427:
426:
424:
411:In 2012 and 2013, Peruvian mathematician
387:
368:
356:
284:
279:
264:
259:
253:
1092:International Congress of Mathematicians
332:≥ 5 is the sum of at most seven primes.
707:
678:
1117:
324:in 1995 showed that every even number
27:Solved conjecture about prime numbers
7:
729:
727:
415:released a pair of papers improving
683:
649:
625:
428:
25:
1379:Conjectures that have been proved
1081:Helfgott, Harald Andres (2014).
717:(Band 1), St.-PĂ©tersbourg 1843,
1374:Conjectures about prime numbers
1010:10.1090/S0025-5718-2013-02733-0
756:"Annals of Mathematics Studies"
633:{\displaystyle {\mathfrak {m}}}
436:{\displaystyle {\mathfrak {M}}}
1083:"The ternary Goldbach problem"
675:
657:
589:{\displaystyle a/q,q<r_{0}}
315:generalized Riemann hypothesis
226:generalized Riemann hypothesis
1:
1124:: CS1 maint: date and year (
917:10.1090/S0025-5718-98-00928-4
872:10.1090/S1079-6762-97-00031-0
169:Some state the conjecture as
163:Annals of Mathematics Studies
1088:. In Jang, Sun Young (ed.).
145:is called "weak" because if
1400:
760:Princeton University Press
616:is a constant. Minor arcs
443:is the union of intervals
234:Ivan Matveevich Vinogradov
224:showed that, assuming the
185:
111:Goldbach's weak conjecture
31:Goldbach's weak conjecture
1192:
939:Kaniecki, Leszek (1995).
35:
1384:Computer-assisted proofs
1187:Prime number conjectures
894:Yannick Saouter (1998).
781:"Harald Andrés Helfgott"
347:In 2002, Liu Ming-Chit (
119:ternary Goldbach problem
1338:Schinzel's hypothesis H
963:10.4064/aa-72-4-361-374
349:University of Hong Kong
115:odd Goldbach conjecture
18:Odd Goldbach conjecture
1369:Analytic number theory
1364:Additive number theory
692:
634:
610:
590:
543:
437:
397:
296:
246:Siegel–Walfisz theorem
232:odd numbers. In 1937,
1343:Waring's prime number
809:"Goldbach Conjecture"
693:
635:
611:
591:
549:around the rationals
544:
438:
398:
297:
188:Goldbach's conjecture
98:Goldbach's conjecture
644:
620:
600:
553:
447:
423:
355:
252:
238:Vinogradov's theorem
113:, also known as the
1308:Legendre's constant
785:webusers.imj-prg.fr
417:major and minor arc
212:Timeline of results
32:
1259:Elliott–Halberstam
1244:Chinese hypothesis
806:Weisstein, Eric W.
688:
640:are defined to be
630:
606:
586:
539:
433:
393:
338:Riemann Hypothesis
292:
242:sufficiently large
230:sufficiently large
194:Christian Goldbach
62:Christian Goldbach
1351:
1350:
1279:Landau's problems
1103:978-89-6105-805-6
994:(286): 997–1038.
609:{\displaystyle c}
103:
102:
16:(Redirected from
1391:
1197:Hardy–Littlewood
1181:
1174:
1167:
1158:
1151:
1150:
1148:
1136:
1130:
1129:
1123:
1115:
1087:
1078:
1072:
1071:
1069:
1057:
1051:
1050:
1048:
1036:
1030:
1029:
1003:
982:
976:
975:
965:
949:Acta Arithmetica
945:
936:
930:
929:
919:
910:(222): 863–866.
900:
891:
885:
884:
874:
856:
847:
841:
840:
838:
826:
820:
819:
818:
801:
795:
794:
792:
791:
777:
771:
770:
768:
767:
752:
746:
745:
743:
731:
722:
712:
697:
695:
694:
689:
687:
686:
674:
669:
664:
653:
652:
639:
637:
636:
631:
629:
628:
615:
613:
612:
607:
595:
593:
592:
587:
585:
584:
563:
548:
546:
545:
540:
538:
534:
527:
522:
521:
503:
486:
481:
480:
462:
442:
440:
439:
434:
432:
431:
402:
400:
399:
394:
392:
391:
373:
372:
301:
299:
298:
293:
291:
290:
289:
288:
271:
270:
269:
268:
123:3-primes problem
40:
33:
21:
1399:
1398:
1394:
1393:
1392:
1390:
1389:
1388:
1354:
1353:
1352:
1347:
1188:
1185:
1155:
1154:
1138:
1137:
1133:
1116:
1104:
1085:
1080:
1079:
1075:
1059:
1058:
1054:
1038:
1037:
1033:
984:
983:
979:
943:
938:
937:
933:
898:
893:
892:
888:
854:
849:
848:
844:
828:
827:
823:
804:
803:
802:
798:
789:
787:
779:
778:
774:
765:
763:
754:
753:
749:
733:
732:
725:
713:
709:
704:
642:
641:
618:
617:
598:
597:
576:
551:
550:
513:
472:
454:
450:
445:
444:
421:
420:
413:Harald Helfgott
383:
364:
353:
352:
334:Leszek Kaniecki
280:
275:
260:
255:
250:
249:
214:
190:
184:
158:Harald Helfgott
80:Harald Helfgott
43:
28:
23:
22:
15:
12:
11:
5:
1397:
1395:
1387:
1386:
1381:
1376:
1371:
1366:
1356:
1355:
1349:
1348:
1346:
1345:
1340:
1335:
1330:
1325:
1320:
1315:
1310:
1305:
1304:
1303:
1298:
1293:
1292:
1291:
1276:
1271:
1266:
1261:
1256:
1251:
1246:
1241:
1236:
1231:
1226:
1221:
1216:
1211:
1210:
1209:
1204:
1193:
1190:
1189:
1186:
1184:
1183:
1176:
1169:
1161:
1153:
1152:
1131:
1102:
1073:
1052:
1031:
977:
956:(4): 361–374.
931:
886:
865:(15): 99–104.
842:
821:
796:
772:
747:
723:
706:
705:
703:
700:
685:
680:
677:
673:
668:
663:
659:
656:
651:
627:
605:
583:
579:
575:
572:
569:
566:
562:
558:
537:
533:
530:
526:
520:
516:
512:
509:
506:
502:
498:
495:
492:
489:
485:
479:
475:
471:
468:
465:
461:
457:
453:
430:
390:
386:
382:
379:
376:
371:
367:
363:
360:
322:Olivier Ramaré
287:
283:
278:
274:
267:
263:
258:
213:
210:
206:
205:
198:Leonhard Euler
186:Main article:
183:
180:
175:
174:
139:
138:
125:, states that
101:
100:
95:
91:
90:
87:
86:First proof in
83:
82:
77:
76:First proof by
73:
72:
69:
68:Conjectured in
65:
64:
59:
58:Conjectured by
55:
54:
49:
45:
44:
41:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1396:
1385:
1382:
1380:
1377:
1375:
1372:
1370:
1367:
1365:
1362:
1361:
1359:
1344:
1341:
1339:
1336:
1334:
1331:
1329:
1326:
1324:
1321:
1319:
1316:
1314:
1311:
1309:
1306:
1302:
1299:
1297:
1294:
1290:
1287:
1286:
1285:
1282:
1281:
1280:
1277:
1275:
1272:
1270:
1267:
1265:
1264:Firoozbakht's
1262:
1260:
1257:
1255:
1252:
1250:
1247:
1245:
1242:
1240:
1237:
1235:
1232:
1230:
1227:
1225:
1222:
1220:
1217:
1215:
1212:
1208:
1205:
1203:
1200:
1199:
1198:
1195:
1194:
1191:
1182:
1177:
1175:
1170:
1168:
1163:
1162:
1159:
1147:
1142:
1135:
1132:
1127:
1121:
1113:
1109:
1105:
1099:
1095:
1093:
1084:
1077:
1074:
1068:
1063:
1056:
1053:
1047:
1042:
1035:
1032:
1027:
1023:
1019:
1015:
1011:
1007:
1002:
997:
993:
990:
989:
981:
978:
973:
969:
964:
959:
955:
951:
950:
942:
935:
932:
927:
923:
918:
913:
909:
906:
905:
897:
890:
887:
882:
878:
873:
868:
864:
860:
853:
846:
843:
837:
832:
825:
822:
816:
815:
810:
807:
800:
797:
786:
782:
776:
773:
761:
757:
751:
748:
742:
737:
730:
728:
724:
720:
716:
711:
708:
701:
699:
666:
654:
603:
581:
577:
573:
570:
567:
564:
560:
556:
535:
531:
528:
524:
518:
514:
510:
507:
504:
500:
496:
493:
490:
487:
483:
477:
473:
469:
466:
463:
459:
455:
451:
418:
414:
409:
406:
388:
384:
380:
377:
374:
369:
365:
361:
358:
350:
345:
343:
339:
335:
331:
327:
323:
319:
316:
312:
308:
303:
285:
281:
276:
272:
265:
261:
256:
247:
243:
239:
235:
231:
227:
223:
219:
211:
209:
203:
202:
201:
199:
195:
189:
181:
179:
172:
171:
170:
167:
165:
164:
159:
154:
152:
150:
144:
136:
132:
128:
127:
126:
124:
120:
116:
112:
108:
107:number theory
99:
96:
92:
88:
84:
81:
78:
74:
70:
66:
63:
60:
56:
53:
52:Number theory
50:
46:
39:
34:
19:
1288:
1229:Bateman–Horn
1134:
1089:
1076:
1055:
1034:
991:
986:
980:
953:
947:
934:
907:
902:
889:
862:
858:
845:
824:
812:
799:
788:. Retrieved
784:
775:
764:. Retrieved
762:. 1996-12-14
759:
750:
714:
710:
410:
346:
329:
325:
320:
309:, Effinger,
307:Deshouillers
304:
215:
207:
191:
176:
168:
161:
155:
148:
140:
122:
118:
114:
110:
104:
1323:Oppermann's
1269:Gilbreath's
1239:Bunyakovsky
1094:Proceedings
988:Math. Comp.
904:Math. Comp.
719:pp. 125–129
342:Terence Tao
340:. In 2012,
240:) that all
147:Goldbach's
1358:Categories
1328:Polignac's
1301:Twin prime
1296:Legendre's
1284:Goldbach's
1214:Agoh–Giuga
1146:1501.05438
836:1501.05438
790:2021-04-06
766:2023-02-05
702:References
222:Littlewood
151:conjecture
143:conjecture
131:odd number
94:Implied by
1313:Lemoine's
1254:Dickson's
1234:Brocard's
1219:Andrica's
1120:cite book
1112:913564239
1067:1305.2897
1046:1205.5252
1001:1201.6656
814:MathWorld
741:1312.7748
679:∖
467:−
381:×
375:≈
305:In 1997,
273:≈
216:In 1923,
156:In 2013,
121:, or the
1318:Mersenne
1249:Cramér's
405:exponent
311:te Riele
1274:Grimm's
1224:Artin's
1026:2618958
1018:3143702
972:1348203
926:1451327
881:1469323
182:Origins
1110:
1100:
1090:Seoul
1024:
1016:
970:
924:
879:
596:where
403:. The
266:16.038
149:strong
135:primes
129:Every
117:, the
1333:PĂłlya
1141:arXiv
1086:(PDF)
1062:arXiv
1041:arXiv
1022:S2CID
996:arXiv
944:(PDF)
899:(PDF)
855:(PDF)
831:arXiv
736:arXiv
218:Hardy
141:This
48:Field
1289:weak
1126:link
1108:OCLC
1098:ISBN
574:<
389:1346
370:3100
362:>
220:and
196:and
89:2013
71:1742
1207:2nd
1202:1st
1006:doi
958:doi
912:doi
867:doi
105:In
1360::
1122:}}
1118:{{
1106:.
1020:.
1014:MR
1012:.
1004:.
992:83
968:MR
966:.
954:72
952:.
946:.
922:MR
920:.
908:67
901:.
877:MR
875:.
861:.
857:.
811:.
783:.
758:.
726:^
698:.
385:10
286:15
109:,
1180:e
1173:t
1166:v
1149:.
1143::
1128:)
1114:.
1070:.
1064::
1049:.
1043::
1028:.
1008::
998::
974:.
960::
928:.
914::
883:.
869::
863:3
839:.
833::
817:.
793:.
769:.
744:.
738::
721:.
684:M
676:)
672:Z
667:/
662:R
658:(
655:=
650:m
626:m
604:c
582:0
578:r
571:q
568:,
565:q
561:/
557:a
536:)
532:x
529:q
525:/
519:0
515:r
511:c
508:+
505:q
501:/
497:a
494:,
491:x
488:q
484:/
478:0
474:r
470:c
464:q
460:/
456:a
452:(
429:M
378:2
366:e
359:n
330:n
326:n
282:3
277:3
262:e
257:e
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
