115:
Romanov initially stated that he had proven the statements "In jedem
Intervall (0, x) liegen mehr als ax Zahlen, welche als Summe von einer Primzahl und einer k-ten Potenz einer ganzen Zahl darstellbar sind, wo a eine gewisse positive, nur von k abhängige Konstante bedeutet" and "In jedem Intervall
69:
406:
116:(0, x) liegen mehr als bx Zahlen, weiche als Summe von einer Primzahl und einer Potenz von a darstellbar sind. Hier ist a eine gegebene ganze Zahl und b eine positive Konstante, welche nur von a abhängt". These statements translate to "In every interval
1094:
464:
522:
1524:
Habsieger, Laurent; Sivak-Fischler, Jimena (2010-12-01). "An effective version of the
Bombieri–Vinogradov theorem, and applications to Chen's theorem and to sums of primes and powers of two".
568:
wrote in 1849 that every odd number larger than 3 can be written as the sum of an odd prime and a power of 2. (He soon noticed a counterexample, namely 959.) This corresponds to the case of
292:
884:
555:
799:
765:
828:
738:
707:
680:
64:
169:
1491:
1376:
922:
252:
193:
623:
280:
229:
146:
1021:
649:
592:
1344:
709:
is shown to be less than 0.5 this implies that the odd numbers that cannot be expressed this way has positive lower asymptotic density.
1121:
by Riegel in 1961. In 2015, the theorem was also proven for polynomials in finite fields. Also in 2015, an arithmetic progression of
411:
469:
1190:
1200:
1658:
401:{\displaystyle d(x)={\frac {\left\vert \{n\leq x:n=p+2^{k},p\ {\textrm {prime,}}\ k\in \mathbb {N} \}\right\vert }{x}}}
286:" respectively. The second statement is generally accepted as the Romanov's theorem, for example in Nathanson's book.
1610:
Shparlinski, Igor E.; Weingartner, Andreas J. (2015-10-30). "An explicit polynomial analogue of
Romanoff's theorem".
1263:
602:, but they were working in the opposite direction, trying to find odd numbers that cannot be expressed in the form.
1663:
1400:
847:
527:
1567:
Rieger, G. J. (1961-02-01). "Verallgemeinerung zweier Sätze von
Romanov aus der additiven Zahlentheorie".
1308:
777:
743:
806:
716:
685:
658:
104:
16:
Theorem on the set of numbers that are the sum of a prime and a positive integer power of the base
95:
is a mathematical theorem proved by
Nikolai Pavlovich Romanov. It states that given a fixed base
1286:
88:
36:
151:
1631:
Madritsch, Manfred G.; Planitzer, Stefan (2018-01-08). "Romanov's
Theorem in Number Fields".
1463:
1348:
894:
234:
178:
1089:{\displaystyle 0.5-{\frac {1}{2^{241}\times 3\times 5\times 7\times 13\times 17\times 241}}}
608:
265:
565:
202:
119:
46:
1219:
Elsholtz, Christian; Schlage-Puchta, Jan-Christoph (2018-04-01). "On
Romanov's constant".
8:
1459:
628:
571:
682:, has been made. The history of such refinements are listed below. In particular, since
1632:
1611:
1592:
1549:
1392:
1244:
1171:
599:
974:; originally proved 0.9367, but an error was found and fixing it would yield 0.093626
1596:
1584:
1553:
1541:
1440:
1325:
1248:
1236:
1196:
1175:
1163:
594:
in the original statement. The counterexample of 959 was, in fact, also mentioned in
1396:
1576:
1533:
1504:
1430:
1317:
1228:
1155:
1122:
99:, the set of numbers that are the sum of a prime and a positive integer power of
1146:
Romanoff, N. P. (1934-12-01). "Über einige Sätze der additiven
Zahlentheorie".
1537:
1435:
1418:
1321:
1303:
1232:
1652:
1588:
1545:
1444:
1329:
1240:
1167:
1118:
1105:
The value cited is 0.4909409303984105956480078184, which is just approximate.
254:
numbers which can be represented as the sum of a prime number and a power of
1580:
1159:
1125:
that are not expressible as the sum of a
Gaussian prime and a power of
1509:
891:; First proof of infinitely many odd numbers that are not of the form
171:
numbers which can be represented as the sum of a prime number and a
1637:
1616:
595:
459:{\displaystyle {\underline {d}}=\liminf _{x\to \infty }d(x)}
605:
In 1934, Romanov proved the theorem. The positive constant
517:{\displaystyle {\overline {d}}=\limsup _{x\to \infty }d(x)}
1117:
Analogous results of
Romanov's theorem has been proven in
195:
is a certain positive constant that is only dependent on
1523:
1609:
1218:
1466:
1458:
Habsieger, Laurent; Roblot, Xavier-Franc¸ois (2006).
1351:
1024:
897:
850:
809:
780:
746:
719:
688:
661:
631:
611:
574:
530:
472:
414:
295:
268:
237:
205:
181:
154:
122:
1630:
1485:
1370:
1088:
958:; Considers only odd numbers; not exact, see note
916:
878:
822:
793:
759:
732:
701:
674:
643:
617:
586:
549:
516:
458:
400:
274:
246:
223:
187:
163:
140:
1650:
655:. Various estimates on the constant, as well as
487:
429:
1457:
1264:"Recherches nouvelles sur les nombres premiers"
1306:(2006-07-01). "A note on Romanov's constant".
1417:Chen, Yong-Gao; Sun, Xue-Gong (2004-06-01).
385:
318:
282:is a positive constant that only depends on
1261:
1192:Additive Number Theory The Classical Bases
1636:
1615:
1508:
1434:
1266:[New research on prime numbers].
1195:. Springer Science & Business Media.
1188:
381:
1145:
1416:
879:{\displaystyle 0.5-5.06\times 10^{-80}}
1651:
1566:
524:. Then Romanov's theorem asserts that
1342:
1302:
550:{\displaystyle {\underline {d}}>0}
1298:
1296:
1214:
1212:
1189:Nathanson, Melvyn B. (2013-03-14).
926:an explicit arithmetic progression
13:
1112:
497:
439:
14:
1675:
1293:
1209:
794:{\displaystyle {\underline {d}}}
760:{\displaystyle {\underline {d}}}
1624:
1603:
1560:
1517:
1451:
1385:Summa Brasiliensis Mathematicae
1099:
823:{\displaystyle {\overline {d}}}
733:{\displaystyle {\overline {d}}}
702:{\displaystyle {\overline {d}}}
675:{\displaystyle {\overline {d}}}
175:-th power of an integer, where
1410:
1336:
1278:
1255:
1182:
1139:
1012:
511:
505:
494:
453:
447:
436:
305:
299:
218:
206:
135:
123:
1:
1132:
87:In mathematics, specifically
815:
725:
694:
667:
478:
110:
7:
10:
1680:
1378:and some related problems"
1309:Acta Mathematica Hungarica
987:Habsieger, Sivak-Fischler
560:
1659:Theorems in number theory
1538:10.1007/s00013-010-0202-5
1460:"On integers of the form
1436:10.1016/j.jnt.2003.11.009
1345:"On Integers of the form
1322:10.1007/s10474-006-0060-6
1233:10.1007/s00209-017-1908-x
1221:Mathematische Zeitschrift
1002:Elsholtz, Schlage-Puchta
199:" and "In every interval
77:
65:Nikolai Pavlovich Romanov
60:
52:
42:
32:
24:
1423:Journal of Number Theory
1419:"On Romanoff's constant"
1262:de Polignac, A. (1849).
164:{\displaystyle \alpha x}
105:lower asymptotic density
1486:{\displaystyle p+2^{k}}
1371:{\displaystyle 2^{k}+p}
917:{\displaystyle 2^{k}+p}
262:is a given integer and
247:{\displaystyle \beta x}
188:{\displaystyle \alpha }
1664:Additive number theory
1487:
1372:
1090:
918:
880:
824:
795:
761:
734:
703:
676:
645:
625:mentioned in the case
619:
618:{\displaystyle \beta }
588:
551:
518:
460:
402:
276:
275:{\displaystyle \beta }
248:
225:
189:
165:
142:
89:additive number theory
37:Additive number theory
1569:Mathematische Annalen
1526:Archiv der Mathematik
1488:
1373:
1148:Mathematische Annalen
1091:
919:
881:
825:
796:
762:
735:
704:
677:
646:
620:
589:
552:
519:
461:
403:
277:
249:
226:
224:{\displaystyle (0,x)}
190:
166:
143:
141:{\displaystyle (0,x)}
1464:
1349:
1343:Erdős, Paul (1950).
1022:
895:
848:
807:
778:
744:
717:
686:
659:
629:
609:
572:
566:Alphonse de Polignac
528:
470:
412:
293:
266:
235:
231:there are more than
203:
179:
152:
148:there are more than
120:
47:Alphonse de Polignac
767:
651:was later known as
644:{\displaystyle a=2}
587:{\displaystyle a=2}
21:
1581:10.1007/BF01396540
1483:
1368:
1287:Letter to Goldbach
1160:10.1007/BF01449161
1086:
955:Habsieger, Roblot
914:
876:
820:
791:
789:
757:
755:
730:
712:
699:
672:
653:Romanov's constant
641:
615:
600:Christian Goldbach
584:
547:
539:
514:
501:
456:
443:
423:
398:
272:
244:
221:
185:
161:
138:
19:
1510:10.4064/aa122-1-4
1202:978-1-4757-3845-2
1123:Gaussian integers
1084:
1008:
1007:
818:
782:
748:
728:
697:
670:
532:
486:
481:
428:
416:
396:
373:
368:
363:
93:Romanov's theorem
85:
84:
20:Romanov's theorem
1671:
1643:
1642:
1640:
1628:
1622:
1621:
1619:
1607:
1601:
1600:
1564:
1558:
1557:
1521:
1515:
1514:
1512:
1497:Acta Arithmetica
1492:
1490:
1489:
1484:
1482:
1481:
1455:
1449:
1448:
1438:
1414:
1408:
1407:
1405:
1399:. Archived from
1382:
1377:
1375:
1374:
1369:
1361:
1360:
1340:
1334:
1333:
1300:
1291:
1282:
1276:
1275:
1259:
1253:
1252:
1216:
1207:
1206:
1186:
1180:
1179:
1143:
1128:
1106:
1103:
1097:
1095:
1093:
1092:
1087:
1085:
1083:
1046:
1045:
1032:
1016:
923:
921:
920:
915:
907:
906:
885:
883:
882:
877:
875:
874:
829:
827:
826:
821:
819:
811:
800:
798:
797:
792:
790:
768:
766:
764:
763:
758:
756:
739:
737:
736:
731:
729:
721:
711:
708:
706:
705:
700:
698:
690:
681:
679:
678:
673:
671:
663:
650:
648:
647:
642:
624:
622:
621:
616:
593:
591:
590:
585:
556:
554:
553:
548:
540:
523:
521:
520:
515:
500:
482:
474:
465:
463:
462:
457:
442:
424:
407:
405:
404:
399:
397:
392:
388:
384:
371:
370:
369:
366:
361:
354:
353:
312:
285:
281:
279:
278:
273:
261:
257:
253:
251:
250:
245:
230:
228:
227:
222:
198:
194:
192:
191:
186:
174:
170:
168:
167:
162:
147:
145:
144:
139:
102:
98:
73:
22:
18:
1679:
1678:
1674:
1673:
1672:
1670:
1669:
1668:
1649:
1648:
1647:
1646:
1629:
1625:
1608:
1604:
1565:
1561:
1522:
1518:
1477:
1473:
1465:
1462:
1461:
1456:
1452:
1415:
1411:
1403:
1380:
1356:
1352:
1350:
1347:
1346:
1341:
1337:
1301:
1294:
1283:
1279:
1260:
1256:
1217:
1210:
1203:
1187:
1183:
1144:
1140:
1135:
1126:
1115:
1113:Generalisations
1110:
1109:
1104:
1100:
1041:
1037:
1036:
1031:
1023:
1020:
1019:
1018:Exact value is
1017:
1013:
925:
902:
898:
896:
893:
892:
867:
863:
849:
846:
845:
810:
808:
805:
804:
803:Upper bound on
781:
779:
776:
775:
774:Lower bound on
747:
745:
742:
741:
720:
718:
715:
714:
713:Refinements of
689:
687:
684:
683:
662:
660:
657:
656:
630:
627:
626:
610:
607:
606:
573:
570:
569:
563:
531:
529:
526:
525:
490:
473:
471:
468:
467:
432:
415:
413:
410:
409:
380:
365:
364:
349:
345:
317:
313:
311:
294:
291:
290:
289:Precisely, let
283:
267:
264:
263:
259:
255:
236:
233:
232:
204:
201:
200:
196:
180:
177:
176:
172:
153:
150:
149:
121:
118:
117:
113:
103:has a positive
100:
96:
67:
17:
12:
11:
5:
1677:
1667:
1666:
1661:
1645:
1644:
1623:
1602:
1559:
1532:(6): 557–566.
1516:
1480:
1476:
1472:
1469:
1450:
1429:(2): 275–284.
1409:
1406:on 2019-02-28.
1367:
1364:
1359:
1355:
1335:
1292:
1277:
1268:Comptes rendus
1254:
1227:(3): 713–724.
1208:
1201:
1181:
1154:(1): 668–678.
1137:
1136:
1134:
1131:
1114:
1111:
1108:
1107:
1098:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1044:
1040:
1035:
1030:
1027:
1010:
1009:
1006:
1005:
1003:
1000:
998:
995:
991:
990:
988:
985:
983:
980:
976:
975:
972:
969:
967:
964:
960:
959:
956:
953:
950:
947:
943:
942:
940:
937:
935:
932:
928:
927:
913:
910:
905:
901:
889:
886:
873:
870:
866:
862:
859:
856:
853:
843:
841:
837:
836:
833:
830:
817:
814:
801:
788:
785:
772:
754:
751:
727:
724:
696:
693:
669:
666:
640:
637:
634:
614:
583:
580:
577:
562:
559:
546:
543:
538:
535:
513:
510:
507:
504:
499:
496:
493:
489:
488:lim sup
485:
480:
477:
455:
452:
449:
446:
441:
438:
435:
431:
430:lim inf
427:
422:
419:
395:
391:
387:
383:
379:
376:
360:
357:
352:
348:
344:
341:
338:
335:
332:
329:
326:
323:
320:
316:
310:
307:
304:
301:
298:
271:
243:
240:
220:
217:
214:
211:
208:
184:
160:
157:
137:
134:
131:
128:
125:
112:
109:
83:
82:
79:
78:First proof in
75:
74:
62:
61:First proof by
58:
57:
54:
53:Conjectured in
50:
49:
44:
43:Conjectured by
40:
39:
34:
30:
29:
26:
15:
9:
6:
4:
3:
2:
1676:
1665:
1662:
1660:
1657:
1656:
1654:
1639:
1634:
1627:
1618:
1613:
1606:
1598:
1594:
1590:
1586:
1582:
1578:
1574:
1571:(in German).
1570:
1563:
1555:
1551:
1547:
1543:
1539:
1535:
1531:
1527:
1520:
1511:
1506:
1502:
1498:
1494:
1478:
1474:
1470:
1467:
1454:
1446:
1442:
1437:
1432:
1428:
1424:
1420:
1413:
1402:
1398:
1394:
1390:
1386:
1379:
1365:
1362:
1357:
1353:
1339:
1331:
1327:
1323:
1319:
1315:
1311:
1310:
1305:
1299:
1297:
1290:. 16-12-1752.
1289:
1288:
1281:
1273:
1270:(in French).
1269:
1265:
1258:
1250:
1246:
1242:
1238:
1234:
1230:
1226:
1222:
1215:
1213:
1204:
1198:
1194:
1193:
1185:
1177:
1173:
1169:
1165:
1161:
1157:
1153:
1150:(in German).
1149:
1142:
1138:
1130:
1124:
1120:
1119:number fields
1102:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1042:
1038:
1033:
1028:
1025:
1015:
1011:
1004:
1001:
999:
996:
993:
992:
989:
986:
984:
981:
978:
977:
973:
970:
968:
965:
962:
961:
957:
954:
951:
948:
945:
944:
941:
938:
936:
933:
930:
929:
911:
908:
903:
899:
890:
887:
871:
868:
864:
860:
857:
854:
851:
844:
842:
839:
838:
834:
831:
812:
802:
786:
783:
773:
770:
769:
752:
749:
722:
710:
691:
664:
654:
638:
635:
632:
612:
603:
601:
598:'s letter to
597:
581:
578:
575:
567:
558:
544:
541:
536:
533:
508:
502:
491:
483:
475:
450:
444:
433:
425:
420:
417:
393:
389:
377:
374:
358:
355:
350:
346:
342:
339:
336:
333:
330:
327:
324:
321:
314:
308:
302:
296:
287:
269:
241:
238:
215:
212:
209:
182:
158:
155:
132:
129:
126:
108:
106:
94:
90:
80:
76:
71:
66:
63:
59:
55:
51:
48:
45:
41:
38:
35:
31:
27:
23:
1626:
1605:
1575:(1): 49–55.
1572:
1568:
1562:
1529:
1525:
1519:
1500:
1496:
1453:
1426:
1422:
1412:
1401:the original
1388:
1384:
1338:
1313:
1307:
1304:Pintz, János
1285:
1280:
1271:
1267:
1257:
1224:
1220:
1191:
1184:
1151:
1147:
1141:
1116:
1101:
1014:
652:
604:
564:
288:
114:
92:
86:
1391:: 113–125.
1316:(1): 1–14.
952:0.49094093
888:Paul Erdős
68: [
1653:Categories
1638:1512.04869
1617:1510.08991
1284:L. Euler,
1274:: 397–401.
1133:References
1129:is given.
982:0.0936275
939:Chen, Xun
1597:121911723
1589:1432-1807
1554:120510181
1546:1420-8938
1503:: 45–50.
1445:0022-314X
1330:1588-2632
1249:125994504
1241:1432-1823
1176:119938116
1168:1432-1807
1078:×
1072:×
1066:×
1060:×
1054:×
1048:×
1029:−
997:0.107648
966:0.093626
869:−
861:×
855:−
816:¯
787:_
753:_
726:¯
695:¯
668:¯
613:β
537:_
498:∞
495:→
479:¯
440:∞
437:→
421:_
378:∈
325:≤
270:β
239:β
183:α
156:α
111:Statement
1397:17379721
408:and let
949:0.0933
934:0.0868
924:through
832:Prover
561:History
258:. Here
28:Theorem
1595:
1587:
1552:
1544:
1443:
1395:
1328:
1247:
1239:
1199:
1174:
1166:
971:Pintz
835:Notes
372:
367:prime,
362:
1633:arXiv
1612:arXiv
1593:S2CID
1550:S2CID
1404:(PDF)
1393:S2CID
1381:(PDF)
1245:S2CID
1172:S2CID
994:2018
979:2010
963:2006
946:2006
931:2004
840:1950
771:Year
596:Euler
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33:Field
1585:ISSN
1542:ISSN
1441:ISSN
1326:ISSN
1237:ISSN
1197:ISBN
1164:ISSN
858:5.06
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542:>
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25:Type
1577:doi
1573:144
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1314:112
1229:doi
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