909:
532:
31:
695:
331:
540:
If it were the case that the sum diverged, then that fact would imply that there are infinitely many twin primes. Because the sum of the reciprocals of the twin primes instead converges, it is not possible to conclude from this result that there are finitely many or infinitely many twin primes.
904:{\displaystyle B_{4}=\left({\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{11}}+{\frac {1}{13}}\right)+\left({\frac {1}{11}}+{\frac {1}{13}}+{\frac {1}{17}}+{\frac {1}{19}}\right)+\left({\frac {1}{101}}+{\frac {1}{103}}+{\frac {1}{107}}+{\frac {1}{109}}\right)+\cdots }
1483:
527:{\displaystyle \sum \limits _{p\,:\,p+2\in \mathbb {P} }{\left({{\frac {1}{p}}+{\frac {1}{p+2}}}\right)}=\left({{\frac {1}{3}}+{\frac {1}{5}}}\right)+\left({{\frac {1}{5}}+{\frac {1}{7}}}\right)+\left({{\frac {1}{11}}+{\frac {1}{13}}}\right)+\cdots }
681:
is a pair of two twin prime pairs, separated by a distance of 4 (the smallest possible distance). The first prime quadruplets are (5, 7, 11, 13), (11, 13, 17, 19), (101, 103, 107, 109). Brun's constant for prime quadruplets, denoted by
320:
That is, twin primes are less frequent than prime numbers by nearly a logarithmic factor. This bound gives the intuition that the sum of the reciprocals of the twin primes converges, or stated in other words, the twin primes form a
315:
1211:
1110:
1328:
1357:
1400:
1003:
1240:
192:
141:
1513:
652:
The last is based on extrapolation from the sum 1.830484424658... for the twin primes below 10. Dominic Klyve showed conditionally (in an unpublished thesis) that
204:
553:
The series converges extremely slowly. Thomas Nicely remarks that after summing the first billion (10) terms, the relative error is still more than 5%.
1012:
80:
1622:"La série 1/5+1/7+1/11+1/13+1/17+1/19+1/29+1/31+1/41+1/43+1/59+1/61+..., où les dénominateurs sont nombres premiers jumeaux est convergente ou finie"
1121:
1824:
1029:
1692:
1657:
924: = 0.87058 83800 ± 0.00000 00005, the error range having a 99% confidence level according to Nicely.
1259:
560:
along the way), Nicely heuristically estimated Brun's constant to be 1.902160578. Nicely has extended his computation to 1.6
1535:
1405:
660:
537:
either has finitely many terms or has infinitely many terms but is convergent: its value is known as Brun's constant.
1649:
1431:
1505:
568:
322:
1804:
1336:
1249:
Many special cases of the above have been proved. Most recently, Jie Wu proved that for sufficiently large
1462:
47:
17:
972:
1819:
1637:
1376:
1219:
572:
1722:
104:
1467:
1712:
1680:
161:
110:
84:
1798:
1461:
Sebah, Pascal; Gourdon, Xavier. "Introduction to twin primes and Brun's constant computation".
1770:
1751:
1688:
1653:
542:
59:
1730:
1578:
1384:
1380:
678:
557:
1703:
Wu, J. (2004) . "Chen's double sieve, Goldbach's conjecture and the twin prime problem".
310:{\displaystyle \pi _{2}(x)=O\!\left({\frac {x(\log \log x)^{2}}{(\log x)^{2}}}\right)\!.}
1726:
1813:
1667:
1436:
Some
Results of Computational Research in Prime Numbers (Computational Number Theory)
39:
103:
The convergence of the sum of reciprocals of twin primes follows from bounds on the
1754:
1641:
931:
144:
92:
55:
1773:
1621:
1543:
1791:
1617:
1601:
1566:
1020:
1016:
564:
10 as of 18 January 2010 but this is not the largest computation of its type.
88:
51:
1206:{\displaystyle \pi _{2}(x)<(2C_{2}+\varepsilon ){\frac {x}{(\log x)^{2}}}}
1778:
1759:
1439:
1379:. Furthermore, academic research on the constant ultimately resulted in the
30:
1367:
The digits of Brun's constant were used in a bid of $ 1,902,160,540 in the
1604:(1915). "Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare".
1787:
1734:
1582:
1372:
1368:
27:
Theorem that the sum of the reciprocals of the twin primes converges
1717:
1388:
1432:"Enumeration to 1.6*10^15 of the twin primes and Brun's constant"
91:
in 1919, and it has historical importance in the introduction of
1105:{\displaystyle \pi _{2}(x)\sim 2C_{2}{\frac {x}{(\log x)^{2}}}.}
1323:{\displaystyle \pi _{2}(x)<4.5\,{\frac {x}{(\log x)^{2}}}}
556:
By calculating the twin primes up to 10 (and discovering the
1799:
Introduction to twin primes and Brun's constant computation
1648:. London Mathematical Society Student Texts. Vol. 66.
1007:
75:
689:, is the sum of the reciprocals of all prime quadruplets:
575:
used all twin primes up to 10 to give the estimate that
1646:
An introduction to sieve methods and their applications
1506:"Dealtalk: Google bid "pi" for Nortel patents and lost"
1401:
Divergence of the sum of the reciprocals of the primes
194:
is the number of twin primes with the smaller at most
1339:
1262:
1222:
1124:
1032:
975:
698:
334:
207:
164:
113:
1484:"Explicit bounds on twin primes and Brun's Constant"
1687:. New York City: Dover Publishing. pp. 1–288.
1351:
1322:
1234:
1205:
1104:
997:
950:. Wolf derived an estimate for the Brun-type sums
903:
526:
309:
186:
135:
1676:Reprinted Providence, RI: Amer. Math. Soc., 1990.
303:
233:
545:only if there are infinitely many twin primes.
928:This constant should not be confused with the
8:
1375:and was one of three Google bids based on
943: + 4), which is also written as
663:). It has been shown unconditionally that
18:Brun's constant for prime quadruplets
1716:
1466:
1338:
1311:
1289:
1288:
1267:
1261:
1221:
1194:
1172:
1157:
1129:
1123:
1090:
1068:
1062:
1037:
1031:
980:
974:
880:
867:
854:
841:
818:
805:
792:
779:
756:
743:
730:
717:
703:
697:
503:
490:
489:
467:
454:
453:
431:
418:
417:
386:
373:
372:
367:
361:
360:
347:
343:
339:
333:
290:
266:
238:
212:
206:
169:
163:
118:
112:
1425:
1423:
1421:
1352:{\displaystyle \varepsilon \approx 3.18}
584:
29:
1801:, 2002. A modern detailed examination.
1606:Archiv for Mathematik og Naturvidenskab
1417:
1371:patent auction. The bid was posted by
1430:Nicely, Thomas R. (18 January 2010).
675:Brun's constant for prime quadruplets
659: < 2.1754 (assuming the
582: ≈ 1.902160583104. Hence,
7:
107:of the sequence of twin primes. Let
1626:Bulletin des Sciences Mathématiques
1567:"Pentium FDIV flaw-lessons learned"
998:{\displaystyle C_{2}=0.6601\ldots }
336:
34:The convergence to Brun's constant.
1797:Sebah, Pascal and Xavier Gourdon,
25:
1235:{\displaystyle \varepsilon >0}
1805:Wolf's article on Brun-type sums
1516:from the original on 3 July 2011
99:Asymptotic bounds on twin primes
1504:Damouni, Nadia (1 July 2011).
1308:
1295:
1279:
1273:
1191:
1178:
1169:
1147:
1141:
1135:
1087:
1074:
1049:
1043:
935:, as prime pairs of the form (
287:
274:
263:
244:
224:
218:
181:
175:
130:
124:
1:
1699:Contains a more modern proof.
1685:Fundamentals of Number Theory
325:. In explicit terms, the sum
1825:Theorems about prime numbers
541:Brun's constant could be an
1674:. Leipzig, Germany: Hirzel.
1242:and all sufficiently large
661:extended Riemann hypothesis
187:{\displaystyle \pi _{2}(x)}
136:{\displaystyle \pi _{2}(x)}
62:to a finite value known as
46:states that the sum of the
1841:
1650:Cambridge University Press
1333:where 4.5 corresponds to
1672:Elementare Zahlentheorie
1406:Meissel–Mertens constant
158:+ 2 is also prime (i.e.
1681:LeVeque, William Judson
1536:"Pentium FDIV flaw FAQ"
670: < 2.347.
1638:Cojocaru, Alina Carmen
1377:mathematical constants
1353:
1324:
1236:
1207:
1106:
999:
905:
528:
311:
188:
137:
83:). Brun's theorem was
35:
1354:
1325:
1237:
1208:
1107:
1000:
906:
529:
312:
189:
143:denote the number of
138:
66:, usually denoted by
33:
1337:
1260:
1220:
1122:
1030:
973:
930:Brun's constant for
696:
332:
205:
162:
111:
1727:2004AcAri.114..215W
1632:: 100–104, 124–128.
1383:becoming a notable
1017:twin prime constant
646:Sebah and Demichel
549:Numerical estimates
58:which differ by 2)
1771:Weisstein, Eric W.
1752:Weisstein, Eric W.
1652:. pp. 73–74.
1565:Price, D. (1995).
1442:on 8 December 2013
1363:In popular culture
1349:
1320:
1232:
1203:
1102:
995:
901:
643:1 × 10
629:1 × 10
615:1 × 10
524:
366:
307:
184:
133:
36:
1755:"Brun's Constant"
1735:10.4064/aa114-3-2
1583:10.1109/40.372360
1318:
1201:
1097:
888:
875:
862:
849:
826:
813:
800:
787:
764:
751:
738:
725:
650:
649:
543:irrational number
511:
498:
475:
462:
439:
426:
402:
381:
335:
297:
198:). Then, we have
16:(Redirected from
1832:
1784:
1783:
1774:"Brun's Theorem"
1765:
1764:
1738:
1720:
1705:Acta Arithmetica
1698:
1675:
1663:
1633:
1613:
1587:
1586:
1562:
1556:
1555:
1553:
1551:
1542:. Archived from
1540:www.trnicely.net
1532:
1526:
1525:
1523:
1521:
1501:
1495:
1494:
1492:
1490:
1482:Klyve, Dominic.
1479:
1473:
1472:
1470:
1458:
1452:
1451:
1449:
1447:
1438:. Archived from
1427:
1385:public relations
1381:Pentium FDIV bug
1358:
1356:
1355:
1350:
1329:
1327:
1326:
1321:
1319:
1317:
1316:
1315:
1290:
1272:
1271:
1241:
1239:
1238:
1233:
1212:
1210:
1209:
1204:
1202:
1200:
1199:
1198:
1173:
1162:
1161:
1134:
1133:
1111:
1109:
1108:
1103:
1098:
1096:
1095:
1094:
1069:
1067:
1066:
1042:
1041:
1010:
1004:
1002:
1001:
996:
985:
984:
910:
908:
907:
902:
894:
890:
889:
881:
876:
868:
863:
855:
850:
842:
832:
828:
827:
819:
814:
806:
801:
793:
788:
780:
770:
766:
765:
757:
752:
744:
739:
731:
726:
718:
708:
707:
679:prime quadruplet
673:There is also a
585:
573:Patrick Demichel
563:
558:Pentium FDIV bug
533:
531:
530:
525:
517:
513:
512:
504:
499:
491:
481:
477:
476:
468:
463:
455:
445:
441:
440:
432:
427:
419:
409:
408:
404:
403:
401:
387:
382:
374:
365:
364:
316:
314:
313:
308:
302:
298:
296:
295:
294:
272:
271:
270:
239:
217:
216:
193:
191:
190:
185:
174:
173:
142:
140:
139:
134:
123:
122:
78:
21:
1840:
1839:
1835:
1834:
1833:
1831:
1830:
1829:
1810:
1809:
1788:Brun's constant
1769:
1768:
1750:
1749:
1746:
1741:
1702:
1695:
1679:
1666:
1660:
1636:
1616:
1600:
1596:
1591:
1590:
1564:
1563:
1559:
1549:
1547:
1546:on 18 June 2019
1534:
1533:
1529:
1519:
1517:
1503:
1502:
1498:
1488:
1486:
1481:
1480:
1476:
1468:10.1.1.464.1118
1460:
1459:
1455:
1445:
1443:
1429:
1428:
1419:
1414:
1397:
1365:
1335:
1334:
1307:
1294:
1263:
1258:
1257:
1218:
1217:
1190:
1177:
1153:
1125:
1120:
1119:
1115:In particular,
1086:
1073:
1058:
1033:
1028:
1027:
1006:
976:
971:
970:
967:
965:Further results
955:
949:
923:
840:
836:
778:
774:
716:
712:
699:
694:
693:
688:
669:
658:
600:
596:
581:
561:
551:
485:
449:
413:
391:
368:
330:
329:
286:
273:
262:
240:
234:
208:
203:
202:
165:
160:
159:
114:
109:
108:
101:
74:
72:
64:Brun's constant
28:
23:
22:
15:
12:
11:
5:
1838:
1836:
1828:
1827:
1822:
1812:
1811:
1808:
1807:
1802:
1795:
1785:
1766:
1745:
1744:External links
1742:
1740:
1739:
1711:(3): 215–273.
1700:
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1614:
1597:
1595:
1592:
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1557:
1527:
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1453:
1416:
1415:
1413:
1410:
1409:
1408:
1403:
1396:
1393:
1364:
1361:
1359:in the above.
1348:
1345:
1342:
1331:
1330:
1314:
1310:
1306:
1303:
1300:
1297:
1293:
1287:
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1146:
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1140:
1137:
1132:
1128:
1113:
1112:
1101:
1093:
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1085:
1082:
1079:
1076:
1072:
1065:
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1057:
1054:
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1040:
1036:
994:
991:
988:
983:
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729:
724:
721:
715:
711:
706:
702:
686:
667:
656:
648:
647:
644:
641:
640:1.902160583104
638:
634:
633:
630:
627:
624:
620:
619:
616:
613:
610:
606:
605:
602:
601:primes below #
597:
594:
589:
579:
550:
547:
535:
534:
523:
520:
516:
510:
507:
502:
497:
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318:
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269:
265:
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211:
183:
180:
177:
172:
168:
132:
129:
126:
121:
117:
100:
97:
70:
44:Brun's theorem
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1837:
1826:
1823:
1821:
1818:
1817:
1815:
1806:
1803:
1800:
1796:
1793:
1789:
1786:
1781:
1780:
1775:
1772:
1767:
1762:
1761:
1756:
1753:
1748:
1747:
1743:
1736:
1732:
1728:
1724:
1719:
1714:
1710:
1706:
1701:
1696:
1694:0-486-68906-9
1690:
1686:
1682:
1678:
1673:
1669:
1665:
1661:
1659:0-521-61275-6
1655:
1651:
1647:
1643:
1642:Murty, M. Ram
1639:
1635:
1631:
1628:(in French).
1627:
1623:
1619:
1615:
1611:
1607:
1603:
1599:
1598:
1593:
1584:
1580:
1576:
1572:
1568:
1561:
1558:
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1528:
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1511:
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1469:
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1457:
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1441:
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1418:
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1402:
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1223:
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1163:
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1154:
1150:
1144:
1138:
1130:
1126:
1118:
1117:
1116:
1099:
1091:
1083:
1080:
1077:
1070:
1063:
1059:
1055:
1052:
1046:
1038:
1034:
1026:
1025:
1024:
1022:
1019:. Then it is
1018:
1014:
1009:
992:
989:
986:
981:
977:
964:
962:
960:
956:
946:
942:
938:
934:
933:
932:cousin primes
920:
917:
916:
915:
898:
895:
891:
885:
882:
877:
872:
869:
864:
859:
856:
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837:
833:
829:
823:
820:
815:
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802:
797:
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789:
784:
781:
775:
771:
767:
761:
758:
753:
748:
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722:
719:
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583:
578:
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570:
565:
559:
554:
548:
546:
544:
538:
521:
518:
514:
508:
505:
500:
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492:
486:
482:
478:
472:
469:
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428:
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410:
405:
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388:
383:
378:
375:
369:
357:
354:
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344:
340:
328:
327:
326:
324:
304:
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267:
259:
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213:
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153:
149:
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127:
119:
115:
106:
98:
96:
94:
93:sieve methods
90:
86:
82:
77:
69:
65:
61:
57:
56:prime numbers
53:
49:
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1550:22 February
1446:16 February
1387:fiasco for
1021:conjectured
626:1.902160578
612:1.902160540
599:set of twin
52:twin primes
48:reciprocals
1814:Categories
1792:PlanetMath
1668:Landau, E.
1594:References
1571:IEEE Micro
1216:for every
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957:of 4/
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