20:
813:
1939:
JĂĄnos Pintz, Approximations to the
Goldbach and twin prime problem and gaps between consecutive primes, Probability and Number Theory (Kanazawa, 2005), Advanced Studies in Pure Mathematics 49, pp. 323â365. Math. Soc. Japan, Tokyo,
250:
303:
658:
1377:
437:
623:
1524:
1411:
371:
1175:
1066:
1431:
650:
567:
488:
showed that there are infinitely many prime pairs with gap bounded by 70 million, and this result has been improved to gaps of length 246 by a collaborative effort of the
1285:
1107:
1330:
1477:
1238:
937:
886:
2711:
2573:
1581:
1211:
997:
179:
847:
1133:
1544:
193:. Bordignon, Johnston, and Starichkova, correcting and improving on Yamada, proved an explicit version of Chen's theorem: every even number greater than
2973:
317:≥ 4 at the cost of using a number which is the product of at most 369 primes rather than a prime or semiprime; under GRH they improve 369 to 33.
1866:
Pintz, Janos (2018). "A new explicit formula in the additive theory of primes with applications II. The exceptional set in
Goldbach's problem".
2624:
de la BretÚche, Régis; Drappeau, Sary (2020). "Niveau de répartition des polynÎmes quadratiques et crible majorant pour les entiers friables".
2138:
2626:
3163:
1788:
Bordignon, Matteo; Starichkova, Valeriia (2022). "An explicit version of Chen's theorem assuming the
Generalized Riemann Hypothesis".
196:
3158:
3018:
1592:
31:
947:.) The existence of infinitely many such primes would follow as a consequence of other number-theoretic conjectures such as the
255:
3001:
2966:
324:
808:{\displaystyle \sum _{\stackrel {p_{n+1}-p_{n}>{\sqrt {p_{n}}}^{1/2}}{p_{n}\leq x}}p_{n+1}-p_{n}\ll x^{0.57+\varepsilon }.}
2996:
1667:
42:. These problems were characterised in his speech as "unattackable at the present state of mathematics" and are now known as
3137:
1913:. Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989). UniversitĂ di Salerno. pp. 115â155.
1809:
Johnston, Daniel R.; Starichkova, Valeriia V. (2022). "Some explicit results on the sum of a prime and an almost prime".
454:
3053:
2959:
493:
2991:
2446:
1244:
320:
2798:
J. Ivanov, Uber die
Primteiler der Zahlen vonder Form A+x^2, Bull. Acad. Sci. St. Petersburg 3 (1895), 361â367.
1746:
Bordignon, Matteo; Johnston, Daniel R.; Starichkova, Valeriia (2022). "An explicit version of Chen's theorem".
1335:
384:
306:
3083:
497:
113:
587:
1547:
1482:
3058:
3023:
1008:
952:
3132:
1966:
2001:
Heath-Brown, D.R.; Puchta, J.-C. (2002). "Integers
Represented as a Sum of Primes and Powers of Two".
1382:
334:
3117:
3063:
3008:
2876:
Baier, Stephan; Zhao, Liangyi (2006). "BombieriâVinogradov type theorems for sparse sets of moduli".
331:
zero, although the set is not proven to be finite. The best current bounds on the exceptional set is
3043:
1138:
252:
is the sum of a prime and a product of at most two primes. Bordignon and
Starichkova reduce this to
3122:
3090:
3078:
1025:
117:
116:, every odd number greater than 5 can be expressed as the sum of three primes, is a consequence of
92: â 1 is a perfect square? In other words: Are there infinitely many primes of the form
74:
50:
1416:
1180:
Merikoski, improving on previous works, showed that there are infinitely many numbers of the form
628:
545:
3107:
3048:
3028:
3013:
2192:
1625:
Merikoski gives two conjectures which would improve the exponent to 1.286 or 1.312, respectively.
1250:
1071:
129:
1293:
3102:
3033:
1646:
Vinogradow, I. M. (November 2002). "Representation of an odd number as a sum of three primes".
1443:
1216:
948:
903:
852:
327:
showed that the exceptional set of even numbers not expressible as the sum of two primes has a
581:
near that size would require a prime gap a hundred million times the size of the average gap.
3095:
3068:
2683:
2545:
1597:
1553:
1183:
969:
900:
Landau's fourth problem asked whether there are infinitely many primes which are of the form
60:
453:
proved that large enough even numbers could be expressed as the sum of two primes and some (
149:
3112:
2895:
2821:
2357:
2322:
2094:
825:
310:
2187:
509:
8:
1112:
3127:
2899:
2825:
2400:
2361:
2326:
3038:
2885:
2781:
2661:
2635:
2586:
2522:
2463:
2381:
2247:
2214:
2168:
2150:
2117:
2098:
2070:
2002:
1983:
1867:
1831:
1810:
1789:
1768:
1747:
1726:
1705:
1684:
1529:
440:
2937:
2934:
2911:
2857:
2852:
2839:
2809:
2665:
2653:
2606:
2526:
2423:
2385:
2373:
2294:
2137:
Alan
Goldston, Daniel; Motohashi, Yoichi; Pintz, Jånos; Yalçın Yıldırım, Cem (2006).
2102:
2039:
1987:
1663:
142:, another weakening of Goldbach's conjecture, proves that for all sufficiently large
139:
2172:
1958:
1888:
2903:
2847:
2829:
2773:
2744:
2645:
2596:
2514:
2455:
2415:
2365:
2330:
2286:
2257:
2160:
2080:
2031:
1975:
1846:
1655:
1651:
489:
2090:
1019:
501:
444:
328:
133:
121:
2035:
584:
JÀrviniemi, improving on Heath-Brown and MatomÀki, shows that there are at most
2481:
2441:
2334:
2235:
1979:
1659:
1004:
1000:
963:
578:
2518:
2348:
Iwaniec, Henryk (1978). "Almost-primes represented by quadratic polynomials".
2085:
2058:
1950:
505:
462:
378:
3152:
2915:
2843:
2657:
2610:
2427:
2377:
2298:
2059:"Variants of the Selberg sieve, and bounded intervals containing many primes"
2043:
959:
819:
466:
78:
35:
23:
2213:
JĂ€rviniemi, Olli (2022). "On large differences between consecutive primes".
1851:
2951:
2861:
2834:
2290:
515:
485:
39:
2262:
2136:
2505:
1616:
A semiprime is a natural number that is the product of two prime factors.
450:
2601:
2541:
2484:(1955). "On a problem in the n-dimensional analytic theory of numbers".
1135:. The best unconditional result is due to Harman and Lewis and it gives
2785:
2749:
2732:
2649:
2467:
2369:
2164:
1954:
1437:
1290:
Baier and Zhao prove that there are infinitely many primes of the form
1012:
570:
523:
2907:
2810:"Using a parity-sensitive sieve to count prime values of a polynomial"
2419:
2313:
Ingham, A. E. (1937). "On the difference between consecutive primes".
2942:
2890:
2277:
Matomaki, K. (2007). "Large differences between consecutive primes".
2155:
2007:
461:
of powers of 2. Following many advances (see Pintz for an overview),
190:
136:
extended this to a full proof of
Goldbach's weak conjecture in 2013.
2777:
2459:
1440:
establishes an upper bound on the density of primes having the form
2640:
2591:
2503:
Harman, G.; Lewis, P. (2001). "Gaussian primes in narrow sectors".
2252:
2219:
1872:
1815:
1794:
1773:
1752:
2122:
2075:
1731:
1725:
Helfgott, H.A. (2013). "The ternary
Goldbach conjecture is true".
1710:
1689:
1911:
On Hardy and
Littlewood's contribution to the Goldbach conjecture
1889:"An Approximate Formula for Goldbach's Problem with Applications"
1240:. Replacing the exponent with 2 would yield Landau's conjecture.
77:: Does there always exist at least one prime between consecutive
54:
2444:(October 1952). "Representations of Primes by Quadratic Forms".
1413:
under the Generalized Riemann Hypothesis for L-functions and to
245:{\displaystyle e^{e^{32,7}}\approx 1.4\cdot 10^{69057979807814}}
19:
2932:
2679:
1745:
1704:
Helfgott, H.A. (2012). "Minor arcs for Goldbach's problem".
1683:
Helfgott, H.A. (2013). "Major arcs for Goldbach's theorem".
944:
1767:
Yamada, Tomohiro (2015-11-11). "Explicit Chen's theorem".
966:
showed that there are infinitely many numbers of the form
298:{\displaystyle e^{e^{15.85}}\approx 3.6\cdot 10^{3321634}}
2764:
Todd, John (1949). "A Problem on Arc Tangent Relations".
2022:
Zhang, Yitang (May 2014). "Bounded gaps between primes".
313:. Johnson and Starichkova give a version working for all
2733:"On the greatest prime factor of a quadratic polynomial"
2623:
2486:
Viliniaus Valst. Univ. Mokslo dardai Chem. Moksly, Ser
2686:
2548:
1959:"On Linnik's approximation to Goldbach's problem. II"
1556:
1532:
1485:
1446:
1419:
1385:
1338:
1296:
1253:
1219:
1186:
1141:
1115:
1074:
1028:
972:
906:
855:
828:
661:
631:
590:
548:
538:
It suffices to check that each prime gap starting at
387:
337:
258:
199:
152:
2401:"Almost-primes represented by quadratic polynomials"
2116:
James, Maynard (2013). "Small gaps between primes".
1787:
1433:
under a certain Elliott-Halberstam type hypothesis.
1808:
530: + 2 is either a prime or a semiprime.
57:
greater than 2 be written as the sum of two primes?
2807:
2705:
2567:
1925:Trudy Matematicheskogo Instituta imeni VA Steklova
1575:
1538:
1518:
1471:
1425:
1405:
1371:
1324:
1279:
1247:shows that infinitely many primes are of the form
1232:
1205:
1169:
1127:
1101:
1060:
991:
931:
880:
841:
807:
644:
617:
561:
496:this was improved to 6, extending earlier work by
431:
365:
297:
244:
173:
2000:
1829:
3150:
625:exceptional primes followed by gaps larger than
16:Four basic unsolved problems about prime numbers
2814:Proceedings of the National Academy of Sciences
2236:"The Differences Between Consecutive Primes, V"
2056:
1022:, there are infinitely many primes of the form
569:. A table of maximal prime gaps shows that the
473:= 8. Assuming the GRH, this can be improved to
1923:Yu V Linnik, Prime numbers and powers of two,
100:As of 2024, all four problems are unresolved.
2967:
518:showed that there are infinitely many primes
2981:
2627:Journal of the European Mathematical Society
2579:Journal of the European Mathematical Society
2398:
943:. (The list of known primes of this form is
103:
2808:Friedlander, John; Iwaniec, Henryk (1997).
2678:Jean-Marc Deshouillers and Henryk Iwaniec,
2502:
2233:
1832:"The exceptional set in Goldbach's problem"
1650:. Series in Pure Mathematics. Vol. 4.
2974:
2960:
2240:International Mathematics Research Notices
2212:
2143:Proceedings of the Japan Academy, Series A
1949:
1830:Montgomery, H. L.; Vaughan, R. C. (1975).
1645:
1109:. Landau's conjecture is for the stronger
2889:
2875:
2851:
2833:
2748:
2639:
2600:
2590:
2539:
2261:
2251:
2218:
2154:
2121:
2084:
2074:
2006:
1871:
1850:
1814:
1793:
1772:
1751:
1730:
1709:
1688:
1372:{\displaystyle a<p^{5/9+\varepsilon }}
432:{\displaystyle E(x)\ll x^{0.5}\log ^{3}x}
2480:
2276:
1908:
1724:
1703:
1682:
1593:List of unsolved problems in mathematics
533:
480:
108:
32:International Congress of Mathematicians
18:
2347:
3151:
2730:
2440:
2312:
1766:
958:One example of near-square primes are
618:{\displaystyle x^{7/100+\varepsilon }}
2955:
2933:
2115:
2063:Research in the Mathematical Sciences
2021:
1865:
1519:{\displaystyle O({\sqrt {x}}/\log x)}
895:
2763:
2279:The Quarterly Journal of Mathematics
1213:with greatest prime factor at least
955:. As of 2024, this problem is open.
822:shows that there is a prime between
2234:Heath-Brown, Roger (October 2020).
63:: Are there infinitely many primes
13:
3164:Unsolved problems in number theory
2185:
1379:; the exponent can be improved to
14:
3175:
2926:
2766:The American Mathematical Monthly
2731:Hooley, Christopher (July 1967).
2139:"Small Gaps between Primes Exist"
1886:
84:Are there infinitely many primes
38:listed four basic problems about
2680:On the greatest prime factor of
2399:Lemke Oliver, Robert J. (2012).
2315:Quarterly Journal of Mathematics
1406:{\displaystyle 1/2+\varepsilon }
999:with at most two prime factors.
366:{\displaystyle E(x)<x^{0.72}}
3159:Conjectures about prime numbers
2869:
2801:
2792:
2757:
2724:
2672:
2617:
2533:
2496:
2474:
2447:American Journal of Mathematics
2434:
2392:
2341:
2306:
2270:
2227:
2206:
2179:
2130:
2109:
2050:
2015:
1994:
1943:
1933:
1917:
1902:
1880:
1859:
1823:
1619:
1802:
1781:
1760:
1739:
1718:
1697:
1676:
1639:
1610:
1513:
1489:
1170:{\displaystyle y=O(p^{0.119})}
1164:
1151:
1096:
1084:
869:
856:
397:
391:
347:
341:
307:Generalized Riemann hypothesis
1:
2716:Annales de l'Institut Fourier
2188:"First occurrence prime gaps"
1632:
1061:{\displaystyle p=x^{2}+y^{2}}
494:ElliottâHalberstam conjecture
1426:{\displaystyle \varepsilon }
645:{\displaystyle {\sqrt {2p}}}
562:{\displaystyle 2{\sqrt {p}}}
7:
2036:10.4007/annals.2014.179.3.7
1586:
1280:{\displaystyle x^{2}+y^{4}}
1245:FriedlanderâIwaniec theorem
1102:{\displaystyle y=O(\log p)}
1009:extended Riemann hypothesis
124:proved it for large enough
10:
3180:
1980:10.1007/s10474-020-01077-8
1967:Acta Mathematica Hungarica
1660:10.1142/9789812776600_0003
1325:{\displaystyle p=an^{2}+1}
1007:proved that, assuming the
114:Goldbach's weak conjecture
71: + 2 is prime?
2987:
2542:"Largest prime factor of
2519:10.1112/S0025579300014388
2086:10.1186/s40687-014-0012-7
1472:{\displaystyle p=n^{2}+1}
1233:{\displaystyle n^{1.279}}
932:{\displaystyle p=n^{2}+1}
881:{\displaystyle (n+1)^{3}}
104:Progress toward solutions
2982:Prime number conjectures
2540:Merikoski, Jori (2022).
2350:Inventiones Mathematicae
2335:10.1093/qmath/os-8.1.255
2057:D.H.J. Polymath (2014).
1603:
492:. Under the generalized
3133:Schinzel's hypothesis H
2706:{\displaystyle n^{2}+1}
2568:{\displaystyle n^{2}+1}
2193:University of Lynchburg
1909:Goldston, D.A. (1992).
1852:10.4064/aa-27-1-353-370
1648:The Goldbach Conjecture
1576:{\displaystyle n^{2}+1}
1206:{\displaystyle n^{2}+1}
992:{\displaystyle n^{2}+1}
953:BatemanâHorn conjecture
888:for every large enough
46:. They are as follows:
2835:10.1073/pnas.94.4.1054
2707:
2569:
1577:
1540:
1520:
1473:
1427:
1407:
1373:
1326:
1281:
1234:
1207:
1171:
1129:
1103:
1062:
993:
949:Bunyakovsky conjecture
933:
882:
843:
809:
646:
619:
563:
433:
367:
299:
246:
175:
174:{\displaystyle 2n=p+q}
27:
26:, German mathematician
3138:Waring's prime number
2708:
2570:
2024:Annals of Mathematics
1578:
1541:
1521:
1474:
1428:
1408:
1374:
1327:
1282:
1235:
1208:
1172:
1130:
1104:
1063:
994:
934:
883:
844:
842:{\displaystyle n^{3}}
810:
647:
620:
564:
534:Legendre's conjecture
481:Twin prime conjecture
434:
368:
311:Dirichlet L-functions
300:
247:
176:
118:Goldbach's conjecture
109:Goldbach's conjecture
75:Legendre's conjecture
61:Twin prime conjecture
51:Goldbach's conjecture
22:
2721::4 (1982), pp. 1â11.
2684:
2546:
2291:10.1093/qmath/ham021
1930:(1951), pp. 152-169.
1554:
1550:numbers of the form
1530:
1483:
1444:
1417:
1383:
1336:
1294:
1251:
1217:
1184:
1139:
1113:
1072:
1026:
970:
904:
853:
826:
659:
629:
588:
546:
385:
335:
256:
197:
150:
130:Vinogradov's theorem
3103:Legendre's constant
2938:"Landau's Problems"
2900:2006AcAri.125..187B
2826:1997PNAS...94.1054F
2362:1978InMat..47..171I
2327:1937QJMat...8..255I
2263:10.1093/imrn/rnz295
2246:(22): 17514â17562.
1128:{\displaystyle y=1}
189:is either prime or
3054:ElliottâHalberstam
3039:Chinese hypothesis
2935:Weisstein, Eric W.
2750:10.1007/BF02395047
2703:
2565:
2370:10.1007/BF01578070
2186:Nicely, Thomas R.
2165:10.3792/pjaa.82.61
1654:. pp. 61â64.
1598:Hilbert's problems
1573:
1536:
1526:such primes up to
1516:
1469:
1423:
1403:
1369:
1322:
1277:
1230:
1203:
1167:
1125:
1099:
1058:
989:
929:
896:Near-square primes
878:
839:
805:
753:
642:
615:
559:
429:
373:(for large enough
363:
295:
242:
171:
28:
3146:
3145:
3074:Landau's problems
2908:10.4064/aa125-2-5
2602:10.4171/jems/1216
2420:10.4064/aa151-3-2
1669:978-981-238-159-0
1539:{\displaystyle x}
1497:
750:
733:
662:
652:; in particular,
640:
557:
469:improved this to
96: + 1?
53:: Can every even
44:Landau's problems
3171:
2992:HardyâLittlewood
2976:
2969:
2962:
2953:
2952:
2948:
2947:
2920:
2919:
2893:
2878:Acta Arithmetica
2873:
2867:
2865:
2855:
2837:
2820:(4): 1054â1058.
2805:
2799:
2796:
2790:
2789:
2761:
2755:
2754:
2752:
2737:Acta Mathematica
2728:
2722:
2712:
2710:
2709:
2704:
2696:
2695:
2676:
2670:
2669:
2650:10.4171/jems/951
2643:
2634:(5): 1577â1624.
2621:
2615:
2614:
2604:
2594:
2585:(4): 1253â1284.
2574:
2572:
2571:
2566:
2558:
2557:
2537:
2531:
2530:
2513:(1â2): 119â135.
2500:
2494:
2493:
2478:
2472:
2471:
2438:
2432:
2431:
2408:Acta Arithmetica
2405:
2396:
2390:
2389:
2345:
2339:
2338:
2310:
2304:
2302:
2274:
2268:
2267:
2265:
2255:
2231:
2225:
2224:
2222:
2210:
2204:
2203:
2201:
2200:
2183:
2177:
2176:
2158:
2134:
2128:
2127:
2125:
2113:
2107:
2106:
2088:
2078:
2054:
2048:
2047:
2030:(3): 1121â1174.
2019:
2013:
2012:
2010:
1998:
1992:
1991:
1963:
1947:
1941:
1937:
1931:
1921:
1915:
1914:
1906:
1900:
1899:
1893:
1884:
1878:
1877:
1875:
1863:
1857:
1856:
1854:
1839:Acta Arithmetica
1836:
1827:
1821:
1820:
1818:
1806:
1800:
1799:
1797:
1785:
1779:
1778:
1776:
1764:
1758:
1757:
1755:
1743:
1737:
1736:
1734:
1722:
1716:
1715:
1713:
1701:
1695:
1694:
1692:
1680:
1674:
1673:
1652:World Scientific
1643:
1626:
1623:
1617:
1614:
1582:
1580:
1579:
1574:
1566:
1565:
1545:
1543:
1542:
1537:
1525:
1523:
1522:
1517:
1503:
1498:
1493:
1478:
1476:
1475:
1470:
1462:
1461:
1432:
1430:
1429:
1424:
1412:
1410:
1409:
1404:
1393:
1378:
1376:
1375:
1370:
1368:
1367:
1357:
1331:
1329:
1328:
1323:
1315:
1314:
1286:
1284:
1283:
1278:
1276:
1275:
1263:
1262:
1239:
1237:
1236:
1231:
1229:
1228:
1212:
1210:
1209:
1204:
1196:
1195:
1176:
1174:
1173:
1168:
1163:
1162:
1134:
1132:
1131:
1126:
1108:
1106:
1105:
1100:
1067:
1065:
1064:
1059:
1057:
1056:
1044:
1043:
1020:Hecke characters
998:
996:
995:
990:
982:
981:
938:
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930:
922:
921:
887:
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846:
845:
840:
838:
837:
818:A result due to
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800:
782:
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769:
768:
752:
751:
749:
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731:
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702:
686:
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651:
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624:
622:
621:
616:
614:
613:
603:
576:
573:holds to 2 â 1.8
568:
566:
565:
560:
558:
553:
542:is smaller than
490:Polymath Project
438:
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296:
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275:
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222:
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180:
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172:
3179:
3178:
3174:
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3172:
3170:
3169:
3168:
3149:
3148:
3147:
3142:
2983:
2980:
2929:
2924:
2923:
2874:
2870:
2806:
2802:
2797:
2793:
2778:10.2307/2305526
2762:
2758:
2729:
2725:
2691:
2687:
2685:
2682:
2681:
2677:
2673:
2622:
2618:
2553:
2549:
2547:
2544:
2543:
2538:
2534:
2501:
2497:
2479:
2475:
2460:10.2307/2372233
2439:
2435:
2403:
2397:
2393:
2346:
2342:
2311:
2307:
2275:
2271:
2232:
2228:
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2207:
2198:
2196:
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2180:
2135:
2131:
2114:
2110:
2055:
2051:
2020:
2016:
1999:
1995:
1961:
1948:
1944:
1938:
1934:
1922:
1918:
1907:
1903:
1891:
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1881:
1864:
1860:
1834:
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1824:
1807:
1803:
1786:
1782:
1765:
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1702:
1698:
1681:
1677:
1670:
1644:
1640:
1635:
1630:
1629:
1624:
1620:
1615:
1611:
1606:
1589:
1583:are composite.
1561:
1557:
1555:
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1528:
1527:
1499:
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1484:
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1214:
1191:
1187:
1185:
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1154:
1140:
1137:
1136:
1114:
1111:
1110:
1073:
1070:
1069:
1052:
1048:
1039:
1035:
1027:
1024:
1023:
977:
973:
971:
968:
967:
917:
913:
905:
902:
901:
898:
872:
868:
854:
851:
850:
833:
829:
827:
824:
823:
790:
786:
777:
773:
758:
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739:
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727:
723:
721:
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632:
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552:
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483:
417:
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386:
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382:
357:
353:
336:
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332:
289:
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268:
264:
263:
259:
257:
254:
253:
236:
232:
209:
205:
204:
200:
198:
195:
194:
151:
148:
147:
134:Harald Helfgott
132:) in 1937, and
122:Ivan Vinogradov
111:
106:
79:perfect squares
17:
12:
11:
5:
3177:
3167:
3166:
3161:
3144:
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3141:
3140:
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3110:
3105:
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3071:
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3021:
3016:
3011:
3006:
3005:
3004:
2999:
2988:
2985:
2984:
2979:
2978:
2971:
2964:
2956:
2950:
2949:
2928:
2927:External links
2925:
2922:
2921:
2884:(2): 187â201.
2868:
2800:
2791:
2772:(8): 517â528.
2756:
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2702:
2699:
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2690:
2671:
2616:
2564:
2561:
2556:
2552:
2532:
2495:
2482:Kubilius, J.P.
2473:
2454:(4): 913â919.
2433:
2414:(3): 241â261.
2391:
2356:(2): 171â188.
2340:
2321:(1): 255â266.
2305:
2285:(4): 489â518.
2269:
2226:
2205:
2178:
2129:
2108:
2049:
2014:
1993:
1974:(2): 569â582.
1942:
1932:
1916:
1901:
1887:Pintz, JĂĄnos.
1879:
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964:Henryk Iwaniec
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579:counterexample
556:
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522:(later called
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140:Chen's theorem
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15:
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3080:
3077:
3076:
3075:
3072:
3070:
3067:
3065:
3062:
3060:
3059:Firoozbakht's
3057:
3055:
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3050:
3047:
3045:
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3040:
3037:
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3032:
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3027:
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2453:
2449:
2448:
2443:
2442:Ankeny, N. C.
2437:
2429:
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2216:
2209:
2195:
2194:
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2182:
2174:
2170:
2166:
2162:
2157:
2152:
2148:
2144:
2140:
2133:
2124:
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2112:
2104:
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2092:
2087:
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2077:
2072:
2068:
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2060:
2053:
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2037:
2033:
2029:
2025:
2018:
2009:
2004:
1997:
1989:
1985:
1981:
1977:
1973:
1969:
1968:
1960:
1957:(July 2020).
1956:
1952:
1946:
1936:
1929:
1926:
1920:
1912:
1905:
1897:
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1622:
1613:
1609:
1599:
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1584:
1570:
1567:
1562:
1558:
1549:
1533:
1510:
1507:
1504:
1500:
1494:
1486:
1466:
1463:
1458:
1454:
1450:
1447:
1439:
1434:
1420:
1400:
1397:
1394:
1390:
1386:
1364:
1361:
1358:
1354:
1350:
1346:
1342:
1339:
1319:
1316:
1311:
1307:
1303:
1300:
1297:
1288:
1272:
1268:
1264:
1259:
1255:
1246:
1241:
1225:
1221:
1200:
1197:
1192:
1188:
1178:
1159:
1155:
1148:
1145:
1142:
1122:
1119:
1116:
1093:
1090:
1087:
1081:
1078:
1075:
1053:
1049:
1045:
1040:
1036:
1032:
1029:
1021:
1017:
1015:
1010:
1006:
1002:
986:
983:
978:
974:
965:
961:
960:Fermat primes
956:
954:
950:
946:
942:
926:
923:
918:
914:
910:
907:
893:
891:
873:
865:
862:
859:
834:
830:
821:
802:
797:
794:
791:
787:
783:
778:
774:
770:
765:
762:
759:
755:
744:
740:
736:
728:
724:
717:
712:
708:
704:
699:
696:
693:
689:
683:
680:
675:
671:
663:
655:
654:
653:
637:
634:
610:
607:
604:
600:
596:
592:
582:
580:
572:
554:
549:
541:
531:
529:
525:
521:
517:
513:
511:
507:
503:
499:
495:
491:
487:
478:
476:
472:
468:
464:
460:
456:
452:
448:
446:
442:
426:
423:
418:
414:
408:
404:
400:
394:
388:
380:
376:
358:
354:
350:
344:
338:
330:
326:
322:
318:
316:
312:
308:
305:assuming the
290:
286:
282:
279:
276:
269:
265:
260:
237:
233:
229:
226:
223:
216:
213:
210:
206:
201:
192:
188:
185:is prime and
184:
168:
165:
162:
159:
156:
153:
145:
141:
137:
135:
131:
127:
123:
119:
115:
101:
95:
91:
87:
83:
80:
76:
73:
70:
66:
62:
59:
56:
52:
49:
48:
47:
45:
41:
40:prime numbers
37:
36:Edmund Landau
33:
25:
24:Edmund Landau
21:
3073:
3024:BatemanâHorn
2941:
2891:math/0602116
2881:
2877:
2871:
2817:
2813:
2803:
2794:
2769:
2765:
2759:
2740:
2736:
2726:
2718:
2715:
2674:
2631:
2625:
2619:
2582:
2578:
2535:
2510:
2504:
2498:
2489:
2485:
2476:
2451:
2445:
2436:
2411:
2407:
2394:
2353:
2349:
2343:
2318:
2314:
2308:
2282:
2278:
2272:
2243:
2239:
2229:
2208:
2197:. Retrieved
2191:
2181:
2156:math/0505300
2149:(4): 61â65.
2146:
2142:
2132:
2111:
2066:
2062:
2052:
2027:
2023:
2017:
2008:math/0201299
1996:
1971:
1965:
1955:Ruzsa, I. Z.
1945:
1935:
1927:
1924:
1919:
1910:
1904:
1895:
1882:
1861:
1842:
1838:
1825:
1804:
1783:
1762:
1741:
1720:
1699:
1678:
1647:
1641:
1621:
1612:
1479:: there are
1435:
1289:
1242:
1179:
1013:
957:
940:
939:for integer
899:
889:
817:
583:
539:
537:
527:
526:) such that
519:
514:
486:Yitang Zhang
484:
474:
470:
458:
449:
374:
319:
314:
186:
182:
143:
138:
125:
112:
99:
93:
89:
85:
68:
64:
43:
30:At the 1912
29:
3118:Oppermann's
3064:Gilbreath's
3034:Bunyakovsky
2743:: 281â299.
2506:Mathematika
1845:: 353â370.
524:Chen primes
457:) constant
455:ineffective
3153:Categories
3123:Polignac's
3096:Twin prime
3091:Legendre's
3079:Goldbach's
3009:AgohâGiuga
2641:1703.03197
2592:1908.08816
2253:1906.09555
2220:2212.10965
2199:2024-09-11
2069:(12): 12.
1873:1804.09084
1816:2208.01229
1795:2211.08844
1774:1511.03409
1753:2207.09452
1633:References
1548:almost all
1438:Brun sieve
1016:-functions
571:conjecture
321:Montgomery
309:(GRH) for
88:such that
67:such that
3108:Lemoine's
3049:Dickson's
3029:Brocard's
3014:Andrica's
2943:MathWorld
2916:0065-1036
2844:0027-8424
2666:146808221
2658:1435-9855
2611:1435-9855
2527:119730332
2428:0065-1036
2386:122656097
2378:0020-9910
2299:0033-5606
2123:1311.4600
2103:119699189
2076:1407.4897
2044:0003-486X
1988:225457520
1951:Pintz, J.
1732:1312.7748
1711:1205.5252
1690:1305.2897
1508:
1421:ε
1401:ε
1365:ε
1091:
798:ε
784:≪
771:−
705:−
681:≤
664:∑
611:ε
443:, due to
424:
401:≪
377:) due to
283:⋅
277:≈
230:⋅
224:≈
191:semiprime
3113:Mersenne
3044:Cramér's
2862:11038598
2173:18847478
1587:See also
1546:. Hence
1005:Kubilius
510:Yıldırım
502:Goldston
445:Goldston
3069:Grimm's
3019:Artin's
2896:Bibcode
2822:Bibcode
2786:2305526
2492:: 5â43.
2468:2372233
2358:Bibcode
2323:Bibcode
2095:3373710
1896:mtak.hu
945:A002496
498:Maynard
329:density
325:Vaughan
291:3321634
55:integer
2914:
2860:
2850:
2842:
2784:
2664:
2656:
2609:
2525:
2466:
2426:
2384:
2376:
2297:
2171:
2101:
2093:
2042:
1986:
1666:
1001:Ankeny
820:Ingham
577:10. A
451:Linnik
439:under
381:, and
181:where
3128:PĂłlya
2886:arXiv
2853:19742
2782:JSTOR
2662:S2CID
2636:arXiv
2587:arXiv
2523:S2CID
2464:JSTOR
2404:(PDF)
2382:S2CID
2248:arXiv
2215:arXiv
2169:S2CID
2151:arXiv
2118:arXiv
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