25:
116:. The direct attack on prime numbers using these methods soon reaches apparently insuperable obstacles, in the way of the accumulation of error terms. In one of the major strands of number theory in the twentieth century, ways were found of avoiding some of the difficulties of a frontal attack with a naive idea of what sieving should be.
3375:
2281:
131:
on these sets (options for giving some elements of these sets more "weight" than others). Furthermore, in some modern applications, sieves are used not to estimate the size of a sifted set, but to produce a function that is large on the set and mostly small outside it, while being easier to analyze
2783:
860:
4043:. Nevertheless, the more advanced sieves can still get very intricate and delicate (especially when combined with other deep techniques in number theory), and entire textbooks have been devoted to this single subfield of number theory; a classic reference is (
1687:
4028:, which roughly speaking asserts that sieve theory methods have extreme difficulty distinguishing between numbers with an odd number of prime factors and numbers with an even number of prime factors. This parity problem is still not very well understood.
565:
1809:
3235:
2048:
1974:
2522:
2886:
437:
3806:. While the original broad aims of sieve theory still are largely unachieved, there have been some partial successes, especially in combination with other number theoretic tools. Highlights include:
2625:
1055:
654:
350:
2053:
731:
949:
3473:
2040:
1472:
1085:
2404:
3516:
3204:
219:
893:
3098:
3060:
3889:
315:
3405:
3022:
1876:
723:
617:
451:
271:
3909:
3692:
3993:
3598:
54:
3670:
3634:
3552:
2617:
2581:
2325:
1852:
1418:
1381:
1344:
3944:
2989:
1307:
1270:
1193:
1156:
986:
3724:
3152:
3125:
2436:
691:
3911:
is sufficiently small (fractions such as 1/10 are quite typical here). This lemma is usually too weak to sieve out primes (which generally require something like
2008:
1706:
1444:
1119:
3370:{\displaystyle \sum \limits _{d\mid n}\mu (d)g(d)=\prod \limits _{\begin{array}{c}p|n;\;p\in \mathbb {P} \end{array}}(1-g(p)),\quad \forall \;n\in \mathbb {N} .}
3768:
3748:
3425:
3227:
2953:
2933:
2913:
2545:
1464:
1233:
1213:
370:
2276:{\displaystyle {\begin{aligned}S({\mathcal {A}},\mathbb {P} ,7)&=A_{1}(x)-A_{2}(x)-A_{3}(x)-A_{5}(x)+A_{6}(x)+A_{10}(x)+A_{15}(x)-A_{30}(x).\end{aligned}}}
123:
numbers), which is typically somewhat larger than the original set, and easier to analyze. More sophisticated sieves also do not work directly with sets
119:
One successful approach is to approximate a specific sifted set of numbers (e.g. the set of prime numbers) by another, simpler set (e.g. the set of
1883:
3843:
even number is the sum of a prime and another number which is either a prime or a semiprime. These can be considered to be near-misses to the
3799:
4519:
4281:
4197:
2452:
317:
we want to sieve. However this abstraction allows for more general situations. Next we introduce a general set of prime numbers called the
2794:
3815:, which shows that the sum of the reciprocals of the twin primes converges (whereas the sum of the reciprocals of all primes diverges);
375:
1088:
4388:
4314:
4243:
4168:
4130:
4100:
3856:
76:
2778:{\displaystyle S({\mathcal {A}},{\mathcal {P}},z)=X\sum \limits _{d\mid P(z)}\mu (d)g(d)+\sum \limits _{d\mid P(z)}\mu (d)r_{d}(x)}
3951:
4379:, Cambridge studies in advanced mathematics, vol. 46, Translated from the second French edition (1995) by C. B. Thomas,
991:
4463:
4331:
4273:
4024:
626:
323:
855:{\displaystyle A_{\operatorname {sift} }:=\{a\in A|(a,p_{1}\cdots p_{k})=1\},\quad p_{1},\dots ,p_{k}\in {\mathcal {P}}}
37:
4458:
47:
41:
33:
898:
4380:
4306:
4092:
4063:
3430:
1682:{\displaystyle |A_{\operatorname {sift} }|=|A|-|E_{2}|-|E_{3}|+|E_{6}|-|E_{5}|+|E_{10}|+|E_{15}|-|E_{30}|+\cdots }
2013:
58:
2958:
The partial sum of the sifting function alternately over- and undercounts, so the remainder term will be huge.
1063:
4022:
The techniques of sieve theory can be quite powerful, but they seem to be limited by an obstacle known as the
2333:
4036:
3802:. One of the original purposes of sieve theory was to try to prove conjectures in number theory such as the
3727:
3478:
3160:
2443:
181:
4070:
to determine efficiently which members of a list of numbers can be completely factored into small primes.
4040:
2892:
868:
4553:
4414:
4336:
4080:
4067:
4055:
3844:
3803:
3065:
3027:
109:
560:{\displaystyle S({\mathcal {A}},{\mathcal {P}},z)=\sum \limits _{n\leq x,{\text{gcd}}(n,P(z))=1}a_{n}.}
3867:
3383:
a word of caution regarding the notation, in the literature one often identifies the set of sequences
276:
4121:, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 72, Berlin:
3386:
2994:
1857:
704:
573:
227:
4192:. London Mathematical Society Monographs. Vol. 33. Princeton, NJ: Princeton University Press.
3848:
4453:
3894:
3675:
4345:
4298:
3958:
3840:
222:
156:
133:
3557:
1815:
4515:
4384:
4310:
4277:
4239:
4227:
4193:
4164:
4126:
4096:
3819:
3811:
3639:
3603:
3521:
2586:
2550:
2294:
1821:
1386:
1349:
1312:
3914:
2965:
1275:
1238:
1161:
1124:
954:
4507:
4423:
4355:
4265:
4223:
4211:
4156:
1804:{\displaystyle S({\mathcal {A}},{\mathcal {P}},z)=\sum \limits _{d\mid P(z)}\mu (d)A_{d}(x)}
4437:
4398:
4367:
4324:
4291:
4253:
4207:
4178:
4140:
4110:
3697:
3130:
3103:
4433:
4394:
4363:
4320:
4287:
4249:
4215:
4203:
4174:
4152:
4136:
4122:
4106:
4059:
2412:
667:
128:
3864:
numbers, then one can accurately estimate the number of elements left in the sieve after
1987:
1423:
1094:
4272:, American Mathematical Society Colloquium Publications, vol. 57, Providence, RI:
4261:
4235:
4035:, in the sense that it does not necessarily require sophisticated concepts from either
3787:
3753:
3733:
3410:
3212:
2938:
2918:
2898:
2530:
1449:
1218:
1198:
355:
113:
4547:
4483:
Brun, Viggo (1915). "Ăśber das
Goldbachsche Gesetz und die Anzahl der Primzahlpaare".
3783:
93:
4405:
4084:
3998:
3836:
3795:
661:
120:
101:
4428:
4409:
4359:
4185:
3791:
620:
160:
4305:, Cambridge Tracts in Mathematics, vol. 70, Cambridge-New York-Melbourne:
4160:
4151:, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 43, Berlin:
3779:
3024:
consisting of restricted Möbius functions. Choosing two appropriate sequences
2959:
148:
4511:
4031:
Compared with other methods in number theory, sieve theory is comparatively
4006:
3832:
1091:. This algorithm works like this: first one removes from the cardinality of
175:
4054:
The sieve methods discussed in this article are not closely related to the
2891:
One tries then to estimate the sifting function by finding upper and lower
1969:{\displaystyle A_{d}(x)=\sum \limits _{n\leq x,n\equiv 0{\pmod {d}}}a_{n}.}
3946:
iterations), but can be enough to obtain results regarding almost primes.
4234:. London Mathematical Society Monographs. Vol. 4. London-New York:
3154:, one can get lower and upper bounds for the original sifting functions
657:
4013:) generalizes Zhang's theorem to arbitrarily long sequences of primes.
2517:{\displaystyle g(1)=1,\qquad 0\leq g(p)<1\qquad p\in \mathbb {P} }
152:
144:
100:
of integers. The prototypical example of a sifted set is the set of
96:, designed to count, or more realistically to estimate the size of,
4350:
3955:, which asserts that there are infinitely many primes of the form
2881:{\displaystyle S({\mathcal {A}},{\mathcal {P}},z)=XG(x,z)+R(x,z).}
432:{\displaystyle P(z)=\prod \limits _{p\in {\mathcal {P}},p<z}p}
108:. Correspondingly, the prototypical example of a sieve is the
151:
in 1915. However Brun's work was inspired by the works of the
18:
2042:. The Möbius function is negative for every prime, so we get
1195:. Now since one has removed the numbers that are divisble by
3484:
3436:
3392:
2816:
2806:
2647:
2637:
2064:
2019:
1863:
1728:
1718:
997:
880:
847:
710:
473:
463:
407:
329:
187:
4091:, London Mathematical Society Student Texts, vol. 66,
3835:(the product of two primes); a closely related theorem of
4377:
Introduction to
Analytic and Probabilistic Number Theory
4504:
4089:
An introduction to sieve methods and their applications
3229:
is multiplicative, one can also work with the identity
1050:{\displaystyle {\mathcal {P}}:=\{2,3,5,7,11,13\dots \}}
127:, but instead count them according to carefully chosen
4303:
Applications of sieve methods to the theory of numbers
3287:
3961:
3917:
3897:
3870:
3756:
3736:
3700:
3678:
3642:
3606:
3560:
3524:
3481:
3433:
3413:
3389:
3238:
3215:
3163:
3133:
3106:
3068:
3030:
2997:
2968:
2941:
2921:
2901:
2797:
2628:
2619:
is some remainder term. The sifting function becomes
2589:
2553:
2533:
2455:
2415:
2336:
2297:
2051:
2016:
1990:
1886:
1860:
1824:
1709:
1475:
1452:
1426:
1389:
1352:
1315:
1278:
1241:
1221:
1201:
1164:
1127:
1097:
1066:
994:
957:
901:
871:
734:
707:
670:
629:
576:
454:
378:
358:
326:
279:
230:
184:
3823:, which shows that there are infinitely many primes
3672:to be already the cardinality of this set. We used
649:{\displaystyle A_{\operatorname {sift} }\subseteq A}
345:{\displaystyle {\mathcal {P}}\subseteq \mathbb {P} }
16:
Ways to estimate the size of sifted sets of integers
1420:, i.e. the cardinality of all numbers divisible by
221:. In the most basic case this sequence is just the
4535:
4066:. Those factorization methods use the idea of the
4048:
3987:
3938:
3903:
3883:
3762:
3742:
3718:
3686:
3664:
3628:
3592:
3546:
3510:
3467:
3419:
3399:
3369:
3221:
3198:
3146:
3119:
3092:
3054:
3016:
2983:
2947:
2927:
2907:
2880:
2777:
2611:
2575:
2539:
2516:
2430:
2398:
2319:
2275:
2034:
2002:
1968:
1870:
1846:
1803:
1681:
1466:. This leads to the inclusion–exclusion principle
1458:
1438:
1412:
1375:
1338:
1301:
1264:
1227:
1207:
1187:
1150:
1113:
1079:
1049:
980:
943:
887:
854:
717:
685:
648:
611:
559:
431:
364:
344:
309:
265:
213:
4119:Lectures on Sieve Methods and Prime Number Theory
4044:
3860:, which asserts that if one is sifting a set of
46:but its sources remain unclear because it lacks
2991:in the sifting function with a weight sequence
4502:Cojocaru, Alina Carmen; Murty, M. Ram (2005).
4005:), which shows that there are infinitely many
944:{\displaystyle E_{p}=\{pn:n\in \mathbb {N} \}}
1060:If one wants to calculate the cardinality of
8:
3468:{\displaystyle {\mathcal {A}}=\{s:s\leq x\}}
3462:
3444:
1044:
1005:
938:
915:
806:
748:
442:The goal of sieve theory is to estimate the
304:
286:
171:For information on notation see at the end.
2035:{\displaystyle {\mathcal {P}}=\mathbb {P} }
163:and only two of his manuscripts survived.
3352:
3304:
1383:again. Additionally one has now to remove
4427:
4349:
4007:pairs of primes within a bounded distance
3979:
3966:
3960:
3926:
3922:
3916:
3896:
3875:
3869:
3755:
3735:
3699:
3680:
3679:
3677:
3647:
3641:
3611:
3605:
3585:
3570:
3561:
3559:
3529:
3523:
3499:
3483:
3482:
3480:
3435:
3434:
3432:
3412:
3391:
3390:
3388:
3360:
3359:
3312:
3311:
3293:
3286:
3243:
3237:
3214:
3187:
3168:
3162:
3138:
3132:
3111:
3105:
3081:
3076:
3067:
3043:
3038:
3029:
3005:
2996:
2967:
2940:
2920:
2900:
2815:
2814:
2805:
2804:
2796:
2760:
2723:
2671:
2646:
2645:
2636:
2635:
2627:
2594:
2588:
2558:
2552:
2532:
2510:
2509:
2454:
2414:
2381:
2341:
2335:
2302:
2296:
2251:
2229:
2207:
2185:
2163:
2141:
2119:
2097:
2073:
2072:
2063:
2062:
2052:
2050:
2028:
2027:
2018:
2017:
2015:
1989:
1957:
1935:
1913:
1891:
1885:
1862:
1861:
1859:
1829:
1823:
1786:
1749:
1727:
1726:
1717:
1716:
1708:
1697:We can rewrite the sifting function with
1668:
1662:
1653:
1645:
1639:
1630:
1622:
1616:
1607:
1599:
1593:
1584:
1576:
1570:
1561:
1553:
1547:
1538:
1530:
1524:
1515:
1507:
1499:
1491:
1485:
1476:
1474:
1451:
1425:
1405:
1399:
1390:
1388:
1368:
1362:
1353:
1351:
1331:
1325:
1316:
1314:
1294:
1288:
1279:
1277:
1257:
1251:
1242:
1240:
1220:
1200:
1180:
1174:
1165:
1163:
1143:
1137:
1128:
1126:
1106:
1098:
1096:
1080:{\displaystyle A_{\operatorname {sift} }}
1071:
1065:
996:
995:
993:
973:
967:
958:
956:
934:
933:
906:
900:
879:
878:
870:
846:
845:
836:
817:
791:
778:
760:
739:
733:
709:
708:
706:
669:
634:
628:
594:
581:
575:
548:
507:
494:
472:
471:
462:
461:
453:
406:
405:
398:
377:
357:
338:
337:
328:
327:
325:
278:
248:
235:
229:
202:
186:
185:
183:
77:Learn how and when to remove this message
3554:is sometimes notated as the cardinality
3100:and denoting the sifting functions with
4475:
4010:
2962:'s idea to improve this was to replace
2399:{\displaystyle A_{d}(x)=g(d)X+r_{d}(x)}
3511:{\displaystyle {\mathcal {A}}=(a_{n})}
3199:{\displaystyle S^{-}\leq S\leq S^{+}.}
1235:twice, one has to add the cardinality
214:{\displaystyle {\mathcal {A}}=(a_{n})}
4334:(2015). "Small gaps between primes".
4002:
7:
3283:
3240:
2720:
2668:
2287:Approximation of the congruence sum
1943:
1936:
1910:
1746:
888:{\displaystyle p\in {\mathcal {P}}}
491:
395:
3349:
3093:{\displaystyle (\lambda _{d}^{+})}
3055:{\displaystyle (\lambda _{d}^{-})}
92:is a set of general techniques in
14:
3857:fundamental lemma of sieve theory
3518:. Also in the literature the sum
697:The inclusion–exclusion principle
178:sequence of non-negative numbers
3884:{\displaystyle N^{\varepsilon }}
3694:to denote the set of primes and
23:
4485:Archiv for Math. Naturvidenskab
3348:
2502:
2477:
1272:. In the next step one removes
812:
310:{\displaystyle A=\{s:s\leq x\}}
4536:Iwaniec & Friedlander 2010
4506:. Cambridge University Press.
4049:Iwaniec & Friedlander 2010
3713:
3701:
3659:
3653:
3623:
3617:
3586:
3582:
3576:
3562:
3541:
3535:
3505:
3492:
3427:itself. This means one writes
3400:{\displaystyle {\mathcal {A}}}
3342:
3339:
3333:
3321:
3294:
3276:
3270:
3264:
3258:
3087:
3069:
3049:
3031:
3017:{\displaystyle (\lambda _{d})}
3011:
2998:
2978:
2972:
2872:
2860:
2851:
2839:
2827:
2801:
2772:
2766:
2753:
2747:
2739:
2733:
2713:
2707:
2701:
2695:
2687:
2681:
2658:
2632:
2606:
2600:
2570:
2564:
2493:
2487:
2465:
2459:
2425:
2419:
2393:
2387:
2368:
2362:
2353:
2347:
2314:
2308:
2263:
2257:
2241:
2235:
2219:
2213:
2197:
2191:
2175:
2169:
2153:
2147:
2131:
2125:
2109:
2103:
2083:
2059:
1947:
1937:
1903:
1897:
1871:{\displaystyle {\mathcal {P}}}
1841:
1835:
1798:
1792:
1779:
1773:
1765:
1759:
1739:
1713:
1669:
1654:
1646:
1631:
1623:
1608:
1600:
1585:
1577:
1562:
1554:
1539:
1531:
1516:
1508:
1500:
1492:
1477:
1406:
1391:
1369:
1354:
1332:
1317:
1295:
1280:
1258:
1243:
1181:
1166:
1144:
1129:
1107:
1099:
974:
959:
797:
765:
761:
718:{\displaystyle {\mathcal {P}}}
680:
674:
612:{\displaystyle a_{n}=1_{A}(n)}
606:
600:
533:
530:
524:
512:
484:
458:
388:
382:
266:{\displaystyle a_{n}=1_{A}(n)}
260:
254:
208:
195:
1:
4410:"Bounded gaps between primes"
4274:American Mathematical Society
4047:) and a more modern text is (
4045:Halberstam & Richert 1974
3800:Goldston-Pintz-Yıldırım sieve
1089:inclusion–exclusion principle
3904:{\displaystyle \varepsilon }
3687:{\displaystyle \mathbb {P} }
988:be the cardinality. Let now
104:up to some prescribed limit
4459:Encyclopedia of Mathematics
4429:10.4007/annals.2014.179.3.7
4360:10.4007/annals.2015.181.1.7
4009:. The Maynard–Tao theorem (
3988:{\displaystyle a^{2}+b^{4}}
3952:Friedlander–Iwaniec theorem
3831:+ 2 is either a prime or a
1854:induced by the elements of
4570:
4381:Cambridge University Press
4375:Tenenbaum, GĂ©rald (1995),
4307:Cambridge University Press
4117:Motohashi, Yoichi (1983),
4093:Cambridge University Press
4064:general number field sieve
4058:sieve methods such as the
4018:Techniques of sieve theory
3778:Modern sieves include the
3593:{\displaystyle |A_{d}(x)|}
4452:Bredikhin, B.M. (2001) ,
4161:10.1007/978-3-662-04658-6
3891:iterations provided that
4512:10.1017/CBO9780511615993
4147:Greaves, George (2001),
3665:{\displaystyle A_{d}(x)}
3636:, while we have defined
3629:{\displaystyle A_{d}(x)}
3547:{\displaystyle A_{d}(x)}
2612:{\displaystyle r_{d}(x)}
2576:{\displaystyle A_{1}(x)}
2320:{\displaystyle A_{d}(x)}
1847:{\displaystyle A_{d}(x)}
1692:
1413:{\displaystyle |E_{30}|}
1376:{\displaystyle |E_{15}|}
1339:{\displaystyle |E_{10}|}
352:and their product up to
32:This article includes a
4149:Sieves in number theory
4037:algebraic number theory
3939:{\displaystyle N^{1/2}}
3728:greatest common divisor
2984:{\displaystyle \mu (d)}
2547:is an approximation of
2444:multiplicative function
1302:{\displaystyle |E_{5}|}
1265:{\displaystyle |E_{6}|}
1188:{\displaystyle |E_{3}|}
1151:{\displaystyle |E_{2}|}
1057:be some set of primes.
981:{\displaystyle |E_{p}|}
134:characteristic function
61:more precise citations.
4190:Prime-detecting sieves
4081:Cojocaru, Alina Carmen
4041:analytic number theory
3989:
3940:
3905:
3885:
3764:
3744:
3720:
3688:
3666:
3630:
3594:
3548:
3512:
3469:
3421:
3401:
3371:
3223:
3200:
3148:
3121:
3094:
3056:
3018:
2985:
2949:
2929:
2909:
2882:
2779:
2613:
2577:
2541:
2518:
2432:
2400:
2321:
2291:One assumes then that
2277:
2036:
2004:
1970:
1872:
1848:
1805:
1683:
1460:
1440:
1414:
1377:
1340:
1303:
1266:
1229:
1209:
1189:
1152:
1115:
1081:
1051:
982:
945:
889:
856:
719:
687:
650:
613:
561:
433:
366:
346:
311:
267:
215:
143:was first used by the
112:, or the more general
4415:Annals of Mathematics
4337:Annals of Mathematics
4068:sieve of Eratosthenes
4056:integer factorization
3990:
3941:
3906:
3886:
3845:twin prime conjecture
3804:twin prime conjecture
3765:
3745:
3721:
3719:{\displaystyle (a,b)}
3689:
3667:
3631:
3595:
3549:
3513:
3475:to define a sequence
3470:
3422:
3402:
3372:
3224:
3201:
3149:
3147:{\displaystyle S^{+}}
3122:
3120:{\displaystyle S^{-}}
3095:
3057:
3019:
2986:
2950:
2930:
2910:
2883:
2780:
2614:
2578:
2542:
2519:
2433:
2401:
2322:
2278:
2037:
2005:
1971:
1873:
1849:
1806:
1684:
1461:
1441:
1415:
1378:
1341:
1304:
1267:
1230:
1210:
1190:
1153:
1116:
1082:
1052:
983:
946:
890:
857:
720:
688:
656:of numbers, that are
651:
619:this just counts the
614:
562:
434:
367:
347:
312:
268:
216:
110:sieve of Eratosthenes
3959:
3915:
3895:
3868:
3754:
3734:
3698:
3676:
3640:
3604:
3558:
3522:
3479:
3431:
3411:
3387:
3236:
3213:
3161:
3131:
3104:
3066:
3028:
2995:
2966:
2939:
2919:
2899:
2795:
2626:
2587:
2551:
2531:
2453:
2431:{\displaystyle g(d)}
2413:
2334:
2295:
2049:
2014:
1988:
1884:
1858:
1822:
1707:
1473:
1450:
1424:
1387:
1350:
1313:
1276:
1239:
1219:
1199:
1162:
1125:
1095:
1087:, one can apply the
1064:
992:
955:
899:
869:
732:
705:
686:{\displaystyle P(z)}
668:
627:
574:
452:
376:
356:
324:
277:
228:
182:
4299:Hooley, Christopher
3849:Goldbach conjecture
3839:asserts that every
3086:
3048:
2003:{\displaystyle z=7}
1818:and some functions
1699:Legendre's identity
1693:Legendre's identity
1439:{\displaystyle 2,3}
1114:{\displaystyle |A|}
865:and for each prime
174:We start with some
4383:, pp. 56–79,
4228:Richert, Hans-Egon
3985:
3936:
3901:
3881:
3841:sufficiently large
3760:
3740:
3716:
3684:
3662:
3626:
3590:
3544:
3508:
3465:
3417:
3397:
3367:
3320:
3318:
3254:
3219:
3196:
3144:
3117:
3090:
3072:
3052:
3034:
3014:
2981:
2945:
2925:
2905:
2878:
2775:
2743:
2691:
2609:
2573:
2537:
2514:
2428:
2396:
2327:can be written as
2317:
2273:
2271:
2032:
2000:
1966:
1952:
1868:
1844:
1801:
1769:
1679:
1456:
1436:
1410:
1373:
1336:
1299:
1262:
1225:
1205:
1185:
1148:
1111:
1077:
1047:
978:
941:
885:
852:
715:
683:
646:
609:
557:
543:
429:
425:
362:
342:
307:
263:
223:indicator function
211:
167:Basic sieve theory
34:list of references
4521:978-0-521-84816-9
4283:978-0-8218-4970-5
4266:Friedlander, John
4224:Halberstam, Heini
4199:978-0-691-12437-7
3763:{\displaystyle b}
3743:{\displaystyle a}
3420:{\displaystyle A}
3282:
3239:
3222:{\displaystyle g}
2948:{\displaystyle R}
2928:{\displaystyle G}
2908:{\displaystyle S}
2719:
2667:
2540:{\displaystyle X}
1909:
1745:
1459:{\displaystyle 5}
1228:{\displaystyle 3}
1208:{\displaystyle 2}
510:
490:
394:
365:{\displaystyle z}
87:
86:
79:
4561:
4539:
4532:
4526:
4525:
4499:
4493:
4492:
4480:
4466:
4441:
4431:
4422:(3): 1121–1174.
4401:
4371:
4353:
4327:
4294:
4257:
4219:
4181:
4143:
4113:
3994:
3992:
3991:
3986:
3984:
3983:
3971:
3970:
3945:
3943:
3942:
3937:
3935:
3934:
3930:
3910:
3908:
3907:
3902:
3890:
3888:
3887:
3882:
3880:
3879:
3774:Types of sieving
3769:
3767:
3766:
3761:
3749:
3747:
3746:
3741:
3725:
3723:
3722:
3717:
3693:
3691:
3690:
3685:
3683:
3671:
3669:
3668:
3663:
3652:
3651:
3635:
3633:
3632:
3627:
3616:
3615:
3599:
3597:
3596:
3591:
3589:
3575:
3574:
3565:
3553:
3551:
3550:
3545:
3534:
3533:
3517:
3515:
3514:
3509:
3504:
3503:
3488:
3487:
3474:
3472:
3471:
3466:
3440:
3439:
3426:
3424:
3423:
3418:
3406:
3404:
3403:
3398:
3396:
3395:
3376:
3374:
3373:
3368:
3363:
3319:
3315:
3297:
3253:
3228:
3226:
3225:
3220:
3205:
3203:
3202:
3197:
3192:
3191:
3173:
3172:
3153:
3151:
3150:
3145:
3143:
3142:
3126:
3124:
3123:
3118:
3116:
3115:
3099:
3097:
3096:
3091:
3085:
3080:
3061:
3059:
3058:
3053:
3047:
3042:
3023:
3021:
3020:
3015:
3010:
3009:
2990:
2988:
2987:
2982:
2954:
2952:
2951:
2946:
2934:
2932:
2931:
2926:
2914:
2912:
2911:
2906:
2887:
2885:
2884:
2879:
2820:
2819:
2810:
2809:
2784:
2782:
2781:
2776:
2765:
2764:
2742:
2690:
2651:
2650:
2641:
2640:
2618:
2616:
2615:
2610:
2599:
2598:
2582:
2580:
2579:
2574:
2563:
2562:
2546:
2544:
2543:
2538:
2523:
2521:
2520:
2515:
2513:
2437:
2435:
2434:
2429:
2405:
2403:
2402:
2397:
2386:
2385:
2346:
2345:
2326:
2324:
2323:
2318:
2307:
2306:
2282:
2280:
2279:
2274:
2272:
2256:
2255:
2234:
2233:
2212:
2211:
2190:
2189:
2168:
2167:
2146:
2145:
2124:
2123:
2102:
2101:
2076:
2068:
2067:
2041:
2039:
2038:
2033:
2031:
2023:
2022:
2009:
2007:
2006:
2001:
1975:
1973:
1972:
1967:
1962:
1961:
1951:
1950:
1896:
1895:
1877:
1875:
1874:
1869:
1867:
1866:
1853:
1851:
1850:
1845:
1834:
1833:
1810:
1808:
1807:
1802:
1791:
1790:
1768:
1732:
1731:
1722:
1721:
1688:
1686:
1685:
1680:
1672:
1667:
1666:
1657:
1649:
1644:
1643:
1634:
1626:
1621:
1620:
1611:
1603:
1598:
1597:
1588:
1580:
1575:
1574:
1565:
1557:
1552:
1551:
1542:
1534:
1529:
1528:
1519:
1511:
1503:
1495:
1490:
1489:
1480:
1465:
1463:
1462:
1457:
1445:
1443:
1442:
1437:
1419:
1417:
1416:
1411:
1409:
1404:
1403:
1394:
1382:
1380:
1379:
1374:
1372:
1367:
1366:
1357:
1345:
1343:
1342:
1337:
1335:
1330:
1329:
1320:
1308:
1306:
1305:
1300:
1298:
1293:
1292:
1283:
1271:
1269:
1268:
1263:
1261:
1256:
1255:
1246:
1234:
1232:
1231:
1226:
1214:
1212:
1211:
1206:
1194:
1192:
1191:
1186:
1184:
1179:
1178:
1169:
1157:
1155:
1154:
1149:
1147:
1142:
1141:
1132:
1121:the cardinality
1120:
1118:
1117:
1112:
1110:
1102:
1086:
1084:
1083:
1078:
1076:
1075:
1056:
1054:
1053:
1048:
1001:
1000:
987:
985:
984:
979:
977:
972:
971:
962:
950:
948:
947:
942:
937:
911:
910:
894:
892:
891:
886:
884:
883:
861:
859:
858:
853:
851:
850:
841:
840:
822:
821:
796:
795:
783:
782:
764:
744:
743:
724:
722:
721:
716:
714:
713:
692:
690:
689:
684:
655:
653:
652:
647:
639:
638:
618:
616:
615:
610:
599:
598:
586:
585:
566:
564:
563:
558:
553:
552:
542:
511:
508:
477:
476:
467:
466:
444:sifting function
438:
436:
435:
430:
424:
411:
410:
371:
369:
368:
363:
351:
349:
348:
343:
341:
333:
332:
316:
314:
313:
308:
272:
270:
269:
264:
253:
252:
240:
239:
220:
218:
217:
212:
207:
206:
191:
190:
159:who died in the
129:weight functions
82:
75:
71:
68:
62:
57:this article by
48:inline citations
27:
26:
19:
4569:
4568:
4564:
4563:
4562:
4560:
4559:
4558:
4544:
4543:
4542:
4533:
4529:
4522:
4501:
4500:
4496:
4482:
4481:
4477:
4473:
4451:
4448:
4404:
4391:
4374:
4330:
4317:
4297:
4284:
4270:Opera de cribro
4262:Iwaniec, Henryk
4260:
4246:
4222:
4200:
4184:
4171:
4153:Springer-Verlag
4146:
4133:
4123:Springer-Verlag
4116:
4103:
4079:
4076:
4060:quadratic sieve
4020:
3975:
3962:
3957:
3956:
3918:
3913:
3912:
3893:
3892:
3871:
3866:
3865:
3776:
3752:
3751:
3732:
3731:
3696:
3695:
3674:
3673:
3643:
3638:
3637:
3607:
3602:
3601:
3566:
3556:
3555:
3525:
3520:
3519:
3495:
3477:
3476:
3429:
3428:
3409:
3408:
3385:
3384:
3317:
3316:
3234:
3233:
3211:
3210:
3183:
3164:
3159:
3158:
3134:
3129:
3128:
3107:
3102:
3101:
3064:
3063:
3026:
3025:
3001:
2993:
2992:
2964:
2963:
2937:
2936:
2917:
2916:
2897:
2896:
2793:
2792:
2756:
2624:
2623:
2590:
2585:
2584:
2554:
2549:
2548:
2529:
2528:
2451:
2450:
2411:
2410:
2377:
2337:
2332:
2331:
2298:
2293:
2292:
2289:
2270:
2269:
2247:
2225:
2203:
2181:
2159:
2137:
2115:
2093:
2086:
2047:
2046:
2012:
2011:
1986:
1985:
1982:
1953:
1887:
1882:
1881:
1856:
1855:
1825:
1820:
1819:
1816:Möbius function
1782:
1705:
1704:
1695:
1658:
1635:
1612:
1589:
1566:
1543:
1520:
1481:
1471:
1470:
1448:
1447:
1422:
1421:
1395:
1385:
1384:
1358:
1348:
1347:
1321:
1311:
1310:
1284:
1274:
1273:
1247:
1237:
1236:
1217:
1216:
1197:
1196:
1170:
1160:
1159:
1133:
1123:
1122:
1093:
1092:
1067:
1062:
1061:
990:
989:
963:
953:
952:
902:
897:
896:
895:denote the set
867:
866:
832:
813:
787:
774:
735:
730:
729:
703:
702:
699:
666:
665:
630:
625:
624:
590:
577:
572:
571:
570:In the case of
544:
450:
449:
374:
373:
354:
353:
322:
321:
275:
274:
244:
231:
226:
225:
198:
180:
179:
169:
83:
72:
66:
63:
52:
38:related reading
28:
24:
17:
12:
11:
5:
4567:
4565:
4557:
4556:
4546:
4545:
4541:
4540:
4527:
4520:
4494:
4474:
4472:
4469:
4468:
4467:
4454:"Sieve method"
4447:
4446:External links
4444:
4443:
4442:
4402:
4389:
4372:
4344:(1): 383–413.
4332:Maynard, James
4328:
4315:
4295:
4282:
4258:
4244:
4236:Academic Press
4220:
4198:
4182:
4169:
4144:
4131:
4114:
4101:
4075:
4072:
4025:parity problem
4019:
4016:
4015:
4014:
3996:
3982:
3978:
3974:
3969:
3965:
3947:
3933:
3929:
3925:
3921:
3900:
3878:
3874:
3852:
3820:Chen's theorem
3816:
3812:Brun's theorem
3775:
3772:
3759:
3739:
3715:
3712:
3709:
3706:
3703:
3682:
3661:
3658:
3655:
3650:
3646:
3625:
3622:
3619:
3614:
3610:
3588:
3584:
3581:
3578:
3573:
3569:
3564:
3543:
3540:
3537:
3532:
3528:
3507:
3502:
3498:
3494:
3491:
3486:
3464:
3461:
3458:
3455:
3452:
3449:
3446:
3443:
3438:
3416:
3394:
3378:
3377:
3366:
3362:
3358:
3355:
3351:
3347:
3344:
3341:
3338:
3335:
3332:
3329:
3326:
3323:
3314:
3310:
3307:
3303:
3300:
3296:
3292:
3289:
3288:
3285:
3281:
3278:
3275:
3272:
3269:
3266:
3263:
3260:
3257:
3252:
3249:
3246:
3242:
3218:
3207:
3206:
3195:
3190:
3186:
3182:
3179:
3176:
3171:
3167:
3141:
3137:
3114:
3110:
3089:
3084:
3079:
3075:
3071:
3051:
3046:
3041:
3037:
3033:
3013:
3008:
3004:
3000:
2980:
2977:
2974:
2971:
2944:
2924:
2904:
2889:
2888:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2826:
2823:
2818:
2813:
2808:
2803:
2800:
2786:
2785:
2774:
2771:
2768:
2763:
2759:
2755:
2752:
2749:
2746:
2741:
2738:
2735:
2732:
2729:
2726:
2722:
2718:
2715:
2712:
2709:
2706:
2703:
2700:
2697:
2694:
2689:
2686:
2683:
2680:
2677:
2674:
2670:
2666:
2663:
2660:
2657:
2654:
2649:
2644:
2639:
2634:
2631:
2608:
2605:
2602:
2597:
2593:
2572:
2569:
2566:
2561:
2557:
2536:
2525:
2524:
2512:
2508:
2505:
2501:
2498:
2495:
2492:
2489:
2486:
2483:
2480:
2476:
2473:
2470:
2467:
2464:
2461:
2458:
2427:
2424:
2421:
2418:
2407:
2406:
2395:
2392:
2389:
2384:
2380:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2355:
2352:
2349:
2344:
2340:
2316:
2313:
2310:
2305:
2301:
2288:
2285:
2284:
2283:
2268:
2265:
2262:
2259:
2254:
2250:
2246:
2243:
2240:
2237:
2232:
2228:
2224:
2221:
2218:
2215:
2210:
2206:
2202:
2199:
2196:
2193:
2188:
2184:
2180:
2177:
2174:
2171:
2166:
2162:
2158:
2155:
2152:
2149:
2144:
2140:
2136:
2133:
2130:
2127:
2122:
2118:
2114:
2111:
2108:
2105:
2100:
2096:
2092:
2089:
2087:
2085:
2082:
2079:
2075:
2071:
2066:
2061:
2058:
2055:
2054:
2030:
2026:
2021:
1999:
1996:
1993:
1981:
1978:
1977:
1976:
1965:
1960:
1956:
1949:
1946:
1942:
1939:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1912:
1908:
1905:
1902:
1899:
1894:
1890:
1865:
1843:
1840:
1837:
1832:
1828:
1812:
1811:
1800:
1797:
1794:
1789:
1785:
1781:
1778:
1775:
1772:
1767:
1764:
1761:
1758:
1755:
1752:
1748:
1744:
1741:
1738:
1735:
1730:
1725:
1720:
1715:
1712:
1694:
1691:
1690:
1689:
1678:
1675:
1671:
1665:
1661:
1656:
1652:
1648:
1642:
1638:
1633:
1629:
1625:
1619:
1615:
1610:
1606:
1602:
1596:
1592:
1587:
1583:
1579:
1573:
1569:
1564:
1560:
1556:
1550:
1546:
1541:
1537:
1533:
1527:
1523:
1518:
1514:
1510:
1506:
1502:
1498:
1494:
1488:
1484:
1479:
1455:
1435:
1432:
1429:
1408:
1402:
1398:
1393:
1371:
1365:
1361:
1356:
1334:
1328:
1324:
1319:
1297:
1291:
1287:
1282:
1260:
1254:
1250:
1245:
1224:
1204:
1183:
1177:
1173:
1168:
1146:
1140:
1136:
1131:
1109:
1105:
1101:
1074:
1070:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
999:
976:
970:
966:
961:
940:
936:
932:
929:
926:
923:
920:
917:
914:
909:
905:
882:
877:
874:
863:
862:
849:
844:
839:
835:
831:
828:
825:
820:
816:
811:
808:
805:
802:
799:
794:
790:
786:
781:
777:
773:
770:
767:
763:
759:
756:
753:
750:
747:
742:
738:
712:
698:
695:
682:
679:
676:
673:
645:
642:
637:
633:
608:
605:
602:
597:
593:
589:
584:
580:
568:
567:
556:
551:
547:
541:
538:
535:
532:
529:
526:
523:
520:
517:
514:
506:
503:
500:
497:
493:
489:
486:
483:
480:
475:
470:
465:
460:
457:
428:
423:
420:
417:
414:
409:
404:
401:
397:
393:
390:
387:
384:
381:
372:as a function
361:
340:
336:
331:
306:
303:
300:
297:
294:
291:
288:
285:
282:
262:
259:
256:
251:
247:
243:
238:
234:
210:
205:
201:
197:
194:
189:
168:
165:
155:mathematician
147:mathematician
114:Legendre sieve
85:
84:
42:external links
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
4566:
4555:
4552:
4551:
4549:
4537:
4531:
4528:
4523:
4517:
4513:
4509:
4505:
4498:
4495:
4490:
4486:
4479:
4476:
4470:
4465:
4461:
4460:
4455:
4450:
4449:
4445:
4439:
4435:
4430:
4425:
4421:
4417:
4416:
4411:
4407:
4406:Zhang, Yitang
4403:
4400:
4396:
4392:
4390:0-521-41261-7
4386:
4382:
4378:
4373:
4369:
4365:
4361:
4357:
4352:
4347:
4343:
4339:
4338:
4333:
4329:
4326:
4322:
4318:
4316:0-521-20915-3
4312:
4308:
4304:
4300:
4296:
4293:
4289:
4285:
4279:
4275:
4271:
4267:
4263:
4259:
4255:
4251:
4247:
4245:0-12-318250-6
4241:
4237:
4233:
4232:Sieve Methods
4229:
4225:
4221:
4217:
4213:
4209:
4205:
4201:
4195:
4191:
4187:
4183:
4180:
4176:
4172:
4170:3-540-41647-1
4166:
4162:
4158:
4154:
4150:
4145:
4142:
4138:
4134:
4132:3-540-12281-8
4128:
4124:
4120:
4115:
4112:
4108:
4104:
4102:0-521-84816-4
4098:
4094:
4090:
4086:
4085:Murty, M. Ram
4082:
4078:
4077:
4073:
4071:
4069:
4065:
4061:
4057:
4052:
4050:
4046:
4042:
4038:
4034:
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4027:
4026:
4017:
4012:
4008:
4004:
4000:
3997:
3980:
3976:
3972:
3967:
3963:
3954:
3953:
3948:
3931:
3927:
3923:
3919:
3898:
3876:
3872:
3863:
3859:
3858:
3853:
3851:respectively.
3850:
3846:
3842:
3838:
3834:
3830:
3826:
3822:
3821:
3817:
3814:
3813:
3809:
3808:
3807:
3805:
3801:
3797:
3793:
3789:
3785:
3784:Selberg sieve
3781:
3773:
3771:
3757:
3737:
3729:
3710:
3707:
3704:
3656:
3648:
3644:
3620:
3612:
3608:
3579:
3571:
3567:
3538:
3530:
3526:
3500:
3496:
3489:
3459:
3456:
3453:
3450:
3447:
3441:
3414:
3407:with the set
3382:
3364:
3356:
3353:
3345:
3336:
3330:
3327:
3324:
3308:
3305:
3301:
3298:
3290:
3279:
3273:
3267:
3261:
3255:
3250:
3247:
3244:
3232:
3231:
3230:
3216:
3193:
3188:
3184:
3180:
3177:
3174:
3169:
3165:
3157:
3156:
3155:
3139:
3135:
3112:
3108:
3082:
3077:
3073:
3044:
3039:
3035:
3006:
3002:
2975:
2969:
2961:
2956:
2942:
2922:
2915:respectively
2902:
2894:
2875:
2869:
2866:
2863:
2857:
2854:
2848:
2845:
2842:
2836:
2833:
2830:
2824:
2821:
2811:
2798:
2791:
2790:
2789:
2769:
2761:
2757:
2750:
2744:
2736:
2730:
2727:
2724:
2716:
2710:
2704:
2698:
2692:
2684:
2678:
2675:
2672:
2664:
2661:
2655:
2652:
2642:
2629:
2622:
2621:
2620:
2603:
2595:
2591:
2567:
2559:
2555:
2534:
2506:
2503:
2499:
2496:
2490:
2484:
2481:
2478:
2474:
2471:
2468:
2462:
2456:
2449:
2448:
2447:
2445:
2441:
2422:
2416:
2390:
2382:
2378:
2374:
2371:
2365:
2359:
2356:
2350:
2342:
2338:
2330:
2329:
2328:
2311:
2303:
2299:
2286:
2266:
2260:
2252:
2248:
2244:
2238:
2230:
2226:
2222:
2216:
2208:
2204:
2200:
2194:
2186:
2182:
2178:
2172:
2164:
2160:
2156:
2150:
2142:
2138:
2134:
2128:
2120:
2116:
2112:
2106:
2098:
2094:
2090:
2088:
2080:
2077:
2069:
2056:
2045:
2044:
2043:
2024:
1997:
1994:
1991:
1979:
1963:
1958:
1954:
1944:
1940:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1906:
1900:
1892:
1888:
1880:
1879:
1878:
1838:
1830:
1826:
1817:
1814:by using the
1795:
1787:
1783:
1776:
1770:
1762:
1756:
1753:
1750:
1742:
1736:
1733:
1723:
1710:
1703:
1702:
1701:
1700:
1676:
1673:
1663:
1659:
1650:
1640:
1636:
1627:
1617:
1613:
1604:
1594:
1590:
1581:
1571:
1567:
1558:
1548:
1544:
1535:
1525:
1521:
1512:
1504:
1496:
1486:
1482:
1469:
1468:
1467:
1453:
1433:
1430:
1427:
1400:
1396:
1363:
1359:
1326:
1322:
1289:
1285:
1252:
1248:
1222:
1202:
1175:
1171:
1138:
1134:
1103:
1090:
1072:
1068:
1058:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1002:
968:
964:
930:
927:
924:
921:
918:
912:
907:
903:
875:
872:
842:
837:
833:
829:
826:
823:
818:
814:
809:
803:
800:
792:
788:
784:
779:
775:
771:
768:
757:
754:
751:
745:
740:
736:
728:
727:
726:
696:
694:
677:
671:
663:
662:prime factors
659:
643:
640:
635:
631:
622:
603:
595:
591:
587:
582:
578:
554:
549:
545:
539:
536:
527:
521:
518:
515:
504:
501:
498:
495:
487:
481:
478:
468:
455:
448:
447:
446:
445:
440:
426:
421:
418:
415:
412:
402:
399:
391:
385:
379:
359:
334:
320:
319:sifting range
301:
298:
295:
292:
289:
283:
280:
257:
249:
245:
241:
236:
232:
224:
203:
199:
192:
177:
172:
166:
164:
162:
158:
154:
150:
146:
142:
137:
135:
130:
126:
122:
117:
115:
111:
107:
103:
102:prime numbers
99:
95:
94:number theory
91:
81:
78:
70:
60:
56:
50:
49:
43:
39:
35:
30:
21:
20:
4554:Sieve theory
4530:
4503:
4497:
4488:
4484:
4478:
4457:
4419:
4413:
4376:
4341:
4335:
4302:
4269:
4231:
4189:
4186:Harman, Glyn
4148:
4118:
4088:
4053:
4032:
4030:
4023:
4021:
4011:Maynard 2015
4001:'s theorem (
3950:
3861:
3855:
3837:Chen Jingrun
3828:
3824:
3818:
3810:
3796:larger sieve
3777:
3600:of some set
3380:
3379:
3208:
2957:
2890:
2788:or in short
2787:
2526:
2442:, meaning a
2439:
2408:
2290:
1983:
1813:
1698:
1696:
1059:
864:
700:
623:of a subset
569:
443:
441:
318:
273:of some set
173:
170:
140:
138:
136:of the set.
124:
121:almost prime
118:
105:
97:
90:Sieve theory
89:
88:
73:
64:
53:Please help
45:
3792:large sieve
3788:Turán sieve
621:cardinality
161:World War I
157:Jean Merlin
98:sifted sets
59:introducing
4471:References
4216:1220.11118
4074:Literature
4033:elementary
4003:Zhang 2014
3827:such that
3780:Brun sieve
2446:such that
149:Viggo Brun
4464:EMS Press
4351:1311.4600
3899:ε
3877:ε
3833:semiprime
3457:≤
3381:Notation:
3357:∈
3350:∀
3328:−
3309:∈
3284:∏
3256:μ
3248:∣
3241:∑
3181:≤
3175:≤
3170:−
3113:−
3074:λ
3045:−
3036:λ
3003:λ
2970:μ
2745:μ
2728:∣
2721:∑
2693:μ
2676:∣
2669:∑
2507:∈
2482:≤
2245:−
2157:−
2135:−
2113:−
1930:≡
1918:≤
1911:∑
1771:μ
1754:∣
1747:∑
1677:⋯
1651:−
1582:−
1536:−
1513:−
1309:and adds
1042:…
931:∈
876:∈
843:∈
827:…
785:⋯
755:∈
641:⊆
499:≤
492:∑
403:∈
396:∏
335:⊆
299:≤
176:countable
145:norwegian
139:The term
132:than the
67:July 2009
4548:Category
4408:(2014).
4301:(1976),
4268:(2010),
4230:(1974).
4188:(2007).
4087:(2006),
4062:and the
3847:and the
3798:and the
3726:for the
951:and let
4438:3171761
4399:1342300
4368:3272929
4325:0404173
4292:2647984
4254:0424730
4208:2331072
4179:1836967
4141:0735437
4111:2200366
2440:density
1980:Example
725:define
660:to the
658:coprime
55:improve
4518:
4436:
4397:
4387:
4366:
4323:
4313:
4290:
4280:
4252:
4242:
4214:
4206:
4196:
4177:
4167:
4139:
4129:
4109:
4099:
3794:, the
3790:, the
3786:, the
3782:, the
3209:Since
2893:bounds
2409:where
153:french
125:per se
4346:arXiv
3999:Zhang
2438:is a
141:sieve
40:, or
4516:ISBN
4385:ISBN
4311:ISBN
4278:ISBN
4240:ISBN
4194:ISBN
4165:ISBN
4127:ISBN
4097:ISBN
3949:The
3854:The
3750:and
3127:and
3062:and
2960:Brun
2935:and
2895:for
2583:and
2527:and
2497:<
2010:and
1984:Let
1487:sift
1446:and
1346:and
1215:and
1158:and
1073:sift
741:sift
701:For
636:sift
419:<
4508:doi
4424:doi
4420:179
4356:doi
4342:181
4212:Zbl
4157:doi
4051:).
4039:or
3730:of
1941:mod
664:of
509:gcd
4550::
4514:.
4489:34
4487:.
4462:,
4456:,
4434:MR
4432:.
4418:.
4412:.
4395:MR
4393:,
4364:MR
4362:.
4354:.
4340:.
4321:MR
4319:,
4309:,
4288:MR
4286:,
4276:,
4264:;
4250:MR
4248:.
4238:.
4226:;
4210:.
4204:MR
4202:.
4175:MR
4173:,
4163:,
4155:,
4137:MR
4135:,
4125:,
4107:MR
4105:,
4095:,
4083:;
3770:.
2955:.
2253:30
2231:15
2209:10
1664:30
1641:15
1618:10
1401:30
1364:15
1327:10
1039:13
1033:11
1003::=
746::=
693:.
439:.
44:,
36:,
4538:)
4534:(
4524:.
4510::
4491:.
4440:.
4426::
4370:.
4358::
4348::
4256:.
4218:.
4159::
3995:.
3981:4
3977:b
3973:+
3968:2
3964:a
3932:2
3928:/
3924:1
3920:N
3873:N
3862:N
3829:p
3825:p
3758:b
3738:a
3714:)
3711:b
3708:,
3705:a
3702:(
3681:P
3660:)
3657:x
3654:(
3649:d
3645:A
3624:)
3621:x
3618:(
3613:d
3609:A
3587:|
3583:)
3580:x
3577:(
3572:d
3568:A
3563:|
3542:)
3539:x
3536:(
3531:d
3527:A
3506:)
3501:n
3497:a
3493:(
3490:=
3485:A
3463:}
3460:x
3454:s
3451::
3448:s
3445:{
3442:=
3437:A
3415:A
3393:A
3365:.
3361:N
3354:n
3346:,
3343:)
3340:)
3337:p
3334:(
3331:g
3325:1
3322:(
3313:P
3306:p
3302:;
3299:n
3295:|
3291:p
3280:=
3277:)
3274:d
3271:(
3268:g
3265:)
3262:d
3259:(
3251:n
3245:d
3217:g
3194:.
3189:+
3185:S
3178:S
3166:S
3140:+
3136:S
3109:S
3088:)
3083:+
3078:d
3070:(
3050:)
3040:d
3032:(
3012:)
3007:d
2999:(
2979:)
2976:d
2973:(
2943:R
2923:G
2903:S
2876:.
2873:)
2870:z
2867:,
2864:x
2861:(
2858:R
2855:+
2852:)
2849:z
2846:,
2843:x
2840:(
2837:G
2834:X
2831:=
2828:)
2825:z
2822:,
2817:P
2812:,
2807:A
2802:(
2799:S
2773:)
2770:x
2767:(
2762:d
2758:r
2754:)
2751:d
2748:(
2740:)
2737:z
2734:(
2731:P
2725:d
2717:+
2714:)
2711:d
2708:(
2705:g
2702:)
2699:d
2696:(
2688:)
2685:z
2682:(
2679:P
2673:d
2665:X
2662:=
2659:)
2656:z
2653:,
2648:P
2643:,
2638:A
2633:(
2630:S
2607:)
2604:x
2601:(
2596:d
2592:r
2571:)
2568:x
2565:(
2560:1
2556:A
2535:X
2511:P
2504:p
2500:1
2494:)
2491:p
2488:(
2485:g
2479:0
2475:,
2472:1
2469:=
2466:)
2463:1
2460:(
2457:g
2426:)
2423:d
2420:(
2417:g
2394:)
2391:x
2388:(
2383:d
2379:r
2375:+
2372:X
2369:)
2366:d
2363:(
2360:g
2357:=
2354:)
2351:x
2348:(
2343:d
2339:A
2315:)
2312:x
2309:(
2304:d
2300:A
2267:.
2264:)
2261:x
2258:(
2249:A
2242:)
2239:x
2236:(
2227:A
2223:+
2220:)
2217:x
2214:(
2205:A
2201:+
2198:)
2195:x
2192:(
2187:6
2183:A
2179:+
2176:)
2173:x
2170:(
2165:5
2161:A
2154:)
2151:x
2148:(
2143:3
2139:A
2132:)
2129:x
2126:(
2121:2
2117:A
2110:)
2107:x
2104:(
2099:1
2095:A
2091:=
2084:)
2081:7
2078:,
2074:P
2070:,
2065:A
2060:(
2057:S
2029:P
2025:=
2020:P
1998:7
1995:=
1992:z
1964:.
1959:n
1955:a
1948:)
1945:d
1938:(
1933:0
1927:n
1924:,
1921:x
1915:n
1907:=
1904:)
1901:x
1898:(
1893:d
1889:A
1864:P
1842:)
1839:x
1836:(
1831:d
1827:A
1799:)
1796:x
1793:(
1788:d
1784:A
1780:)
1777:d
1774:(
1766:)
1763:z
1760:(
1757:P
1751:d
1743:=
1740:)
1737:z
1734:,
1729:P
1724:,
1719:A
1714:(
1711:S
1674:+
1670:|
1660:E
1655:|
1647:|
1637:E
1632:|
1628:+
1624:|
1614:E
1609:|
1605:+
1601:|
1595:5
1591:E
1586:|
1578:|
1572:6
1568:E
1563:|
1559:+
1555:|
1549:3
1545:E
1540:|
1532:|
1526:2
1522:E
1517:|
1509:|
1505:A
1501:|
1497:=
1493:|
1483:A
1478:|
1454:5
1434:3
1431:,
1428:2
1407:|
1397:E
1392:|
1370:|
1360:E
1355:|
1333:|
1323:E
1318:|
1296:|
1290:5
1286:E
1281:|
1259:|
1253:6
1249:E
1244:|
1223:3
1203:2
1182:|
1176:3
1172:E
1167:|
1145:|
1139:2
1135:E
1130:|
1108:|
1104:A
1100:|
1069:A
1045:}
1036:,
1030:,
1027:7
1024:,
1021:5
1018:,
1015:3
1012:,
1009:2
1006:{
998:P
975:|
969:p
965:E
960:|
939:}
935:N
928:n
925::
922:n
919:p
916:{
913:=
908:p
904:E
881:P
873:p
848:P
838:k
834:p
830:,
824:,
819:1
815:p
810:,
807:}
804:1
801:=
798:)
793:k
789:p
780:1
776:p
772:,
769:a
766:(
762:|
758:A
752:a
749:{
737:A
711:P
681:)
678:z
675:(
672:P
644:A
632:A
607:)
604:n
601:(
596:A
592:1
588:=
583:n
579:a
555:.
550:n
546:a
540:1
537:=
534:)
531:)
528:z
525:(
522:P
519:,
516:n
513:(
505:,
502:x
496:n
488:=
485:)
482:z
479:,
474:P
469:,
464:A
459:(
456:S
427:p
422:z
416:p
413:,
408:P
400:p
392:=
389:)
386:z
383:(
380:P
360:z
339:P
330:P
305:}
302:x
296:s
293::
290:s
287:{
284:=
281:A
261:)
258:n
255:(
250:A
246:1
242:=
237:n
233:a
209:)
204:n
200:a
196:(
193:=
188:A
106:X
80:)
74:(
69:)
65:(
51:.
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