555:
It is reasonable to allow negative numbers to count as primes as a step towards formulating more general conjectures that apply to other systems of numbers than the integers, but at the same time it is easy to just negate the polynomials if necessary to reduce to the case where the leading
847:
259:
543:
produces only negative numbers when given a positive argument, so the fraction of prime numbers among its values is always zero. There are two equally valid ways of refining the conjecture to avoid this difficulty:
675:
376:
468:
1153:
123:
or there would be a polynomial with infinitely many roots, whereas the conjecture is how to give conditions where the values are simultaneously prime for infinitely many
939:
506:
408:
64:
The
Bateman–Horn conjecture provides a conjectured density for the positive integers at which a given set of polynomials all have prime values. For a set of
87:
with integer coefficients, an obvious necessary condition for the polynomials to simultaneously generate prime values infinitely often is that they satisfy
1343:
167:
1338:
1198:
1181:
1146:
1176:
41:
1068:
842:{\displaystyle \pi _{2}(x)\sim 2\prod _{p\geq 3}{\frac {p(p-2)}{(p-1)^{2}}}{\frac {x}{(\log x)^{2}}}\approx 1.32{\frac {x}{(\log x)^{2}}}.}
548:
One may require all the polynomials to have positive leading coefficients, so that only a constant number of their values can be negative.
520:
is itself positive, and with some work this can be proved. (Work is needed since some infinite products of positive numbers equal zero.)
1317:
551:
Alternatively, one may allow negative leading coefficients but count a negative number as being prime when its absolute value is prime.
1233:
1139:
281:
1171:
1348:
1263:
1006:
Bateman, Paul T.; Horn, Roger A. (1962), "A heuristic asymptotic formula concerning the distribution of prime numbers",
976:
runs over nonconstant polynomials with degree that is a multiple of 4. An analogue of the
Bateman–Horn conjecture over
1238:
1312:
53:
1188:
1297:
1243:
88:
1223:
416:
1302:
1270:
1258:
1287:
1228:
1208:
1193:
1282:
1213:
69:
964:
of odd degree, but it appears to take (asymptotically) twice as many irreducible values as expected when
1275:
1248:
1292:
1253:
1094:
980:
which fits numerical data uses an additional factor in the asymptotics which depends on the value of
595:
1085:
Friedlander, John; Granville, Andrew (1991), "Limitations to the equi-distribution of primes. IV.",
1307:
1218:
1119:
1025:
910:
476:
1064:
594:) is prime are themselves the prime numbers, and the conjecture becomes a restatement of the
1102:
1074:
1041:
1015:
1037:
665:
on the density of twin primes, according to which the number of twin prime pairs less than
1078:
1060:
1045:
1033:
384:
33:
1098:
1052:
40:
who proposed it in 1962. It provides a vast generalization of such conjectures as the
1332:
1114:
Soren Laing
Alethia-Zomlefer; Lenny Fukshansky; Stephan Ramon Garcia (25 July 2018),
17:
157:) is the number of prime-generating integers among the positive integers less than
25:
254:{\displaystyle P(x)\sim {\frac {C}{D}}\int _{2}^{x}{\frac {dt}{(\log t)^{m}}},\,}
662:
658:
45:
37:
29:
902:, but the analogue is wrong. For example, data suggest that the polynomial
134:
is prime-generating for the given system of polynomials if every polynomial
1131:
1106:
968:
runs over polynomials of degree that is 2 mod 4, while it (provably) takes
111:, having all values of the polynomials simultaneously prime for a given
1029:
953:
takes (asymptotically) the expected number of irreducible values when
516:
is positive. Intuitively one then naturally expects that the constant
1020:
657:) are prime are just the smaller of the two primes in every pair of
1124:
528:
As stated above, the conjecture is not true: the single polynomial
1135:
564:
If the system of polynomials consists of the single polynomial
601:
If the system of polynomials consists of the two polynomials
371:{\displaystyle C=\prod _{p}{\frac {1-N(p)/p}{(1-1/p)^{m}}}\ }
661:. In this case, the Bateman–Horn conjecture reduces to the
268:
is the product of the degrees of the polynomials and where
898:
suggest an analogue of the
Bateman–Horn conjecture over
115:
would imply that at least one of them must be equal to
857:
When the integers are replaced by the polynomial ring
913:
678:
479:
419:
387:
284:
170:
865:, one can ask how often a finite set of polynomials
119:, which can only happen for finitely many values of
933:
841:
500:
462:
402:
370:
253:
161:, then the Bateman–Horn conjecture states that
52: + 1; it is also a strengthening of
1147:
1116:ONE CONJECTURE TO RULE THEM ALL: BATEMAN–HORN
894:. Well-known analogies between integers and
8:
882:simultaneously takes irreducible values in
853:Analogue for polynomials over a finite field
91:, that there does not exist a prime number
24:is a statement concerning the frequency of
1154:
1140:
1132:
463:{\displaystyle f(n)\equiv 0{\pmod {p}}.\ }
48:or their conjecture on primes of the form
1123:
1019:
930:
918:
912:
827:
805:
790:
768:
759:
720:
708:
683:
677:
512:, so each factor in the infinite product
478:
438:
418:
386:
356:
344:
322:
301:
295:
283:
250:
238:
211:
205:
200:
186:
169:
145:) produces a prime number when given
7:
988:is the degree of the polynomials in
631: + 2, then the values of
446:
1344:Unsolved problems in number theory
1087:Proceedings of the Royal Society A
1057:Unsolved problems in number theory
107:. For, if there were such a prime
14:
54:Schinzel's hypothesis H
473:Bunyakovsky's property implies
28:among the values of a system of
1339:Conjectures about prime numbers
972:irreducible values at all when
439:
42:Hardy and Littlewood conjecture
824:
811:
787:
774:
756:
743:
738:
726:
695:
689:
489:
483:
450:
440:
429:
423:
397:
391:
353:
332:
319:
313:
235:
222:
180:
174:
1:
103:) for every positive integer
32:, named after mathematicians
95:that divides their product
663:Hardy–Littlewood conjecture
556:coefficients are positive.
410:the number of solutions to
272:is the product over primes
1365:
1008:Mathematics of Computation
1167:
957:runs over polynomials in
934:{\displaystyle x^{3}+u\,}
501:{\displaystyle N(p)<p}
1162:Prime number conjectures
1313:Schinzel's hypothesis H
886:when we substitute for
70:irreducible polynomials
22:Bateman–Horn conjecture
1349:Analytic number theory
1107:10.1098/rspa.1991.0138
935:
843:
502:
464:
404:
372:
255:
89:Bunyakovsky's property
1318:Waring's prime number
936:
844:
539:) = −
503:
465:
405:
373:
256:
911:
676:
596:prime number theorem
477:
417:
403:{\displaystyle N(p)}
385:
282:
168:
149:as its argument. If
1283:Legendre's constant
1099:1991RSPSA.435..197F
861:for a finite field
210:
1234:Elliott–Halberstam
1219:Chinese hypothesis
931:
839:
719:
579:, then the values
498:
460:
400:
368:
300:
251:
196:
44:on the density of
1326:
1325:
1254:Landau's problems
1118:, pp. 1–45,
1093:(1893): 197–204,
1070:978-0-387-20860-2
834:
797:
766:
704:
459:
367:
363:
291:
245:
194:
78:, ...,
1356:
1172:Hardy–Littlewood
1156:
1149:
1142:
1133:
1128:
1127:
1109:
1081:
1059:(3rd ed.),
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741:
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718:
688:
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524:Negative numbers
507:
505:
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1353:
1329:
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1322:
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1113:
1084:
1071:
1061:Springer-Verlag
1053:Guy, Richard K.
1051:
1021:10.2307/2004056
1014:(79): 363–367,
1005:
1002:
963:
952:
914:
909:
908:
873:
855:
823:
810:
786:
773:
755:
742:
722:
679:
674:
673:
652:
641:
635:for which both
622:
607:
589:
570:
562:
534:
526:
508:for all primes
475:
474:
415:
414:
383:
382:
352:
331:
303:
280:
279:
234:
221:
213:
166:
165:
139:
86:
77:
62:
34:Paul T. Bateman
12:
11:
5:
1362:
1360:
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1323:
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1300:
1295:
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1226:
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1216:
1211:
1206:
1201:
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1191:
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1168:
1165:
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1144:
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1111:
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1069:
1049:
1001:
998:
961:
950:
944:
943:
942:
941:
929:
926:
921:
917:
869:
854:
851:
850:
849:
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830:
826:
822:
819:
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813:
809:
804:
801:
793:
789:
785:
782:
779:
776:
772:
762:
758:
754:
751:
748:
745:
740:
737:
734:
731:
728:
725:
717:
714:
711:
707:
703:
700:
697:
694:
691:
686:
682:
650:
639:
627:) =
620:
612:) =
605:
587:
575:) =
568:
561:
558:
553:
552:
549:
532:
525:
522:
497:
494:
491:
488:
485:
482:
471:
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456:
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437:
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298:
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262:
261:
249:
241:
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208:
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199:
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185:
182:
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137:
82:
75:
61:
58:
13:
10:
9:
6:
4:
3:
2:
1361:
1350:
1347:
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1337:
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1316:
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1299:
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1277:
1274:
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1262:
1261:
1260:
1257:
1256:
1255:
1252:
1250:
1247:
1245:
1242:
1240:
1239:Firoozbakht's
1237:
1235:
1232:
1230:
1227:
1225:
1222:
1220:
1217:
1215:
1212:
1210:
1207:
1205:
1202:
1200:
1197:
1195:
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1190:
1187:
1183:
1180:
1178:
1175:
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1170:
1169:
1166:
1157:
1152:
1150:
1145:
1143:
1138:
1137:
1134:
1126:
1121:
1117:
1112:
1108:
1104:
1100:
1096:
1092:
1088:
1083:
1080:
1076:
1072:
1066:
1062:
1058:
1054:
1050:
1047:
1043:
1039:
1035:
1031:
1027:
1022:
1017:
1013:
1009:
1004:
1003:
999:
997:
995:
991:
987:
984:mod 4, where
983:
979:
975:
971:
967:
960:
956:
949:
927:
924:
919:
915:
907:
906:
905:
904:
903:
901:
897:
893:
889:
885:
881:
877:
872:
868:
864:
860:
852:
836:
828:
820:
817:
814:
807:
802:
799:
791:
783:
780:
777:
770:
760:
752:
749:
746:
735:
732:
729:
723:
715:
712:
709:
705:
701:
698:
692:
684:
680:
672:
671:
670:
668:
664:
660:
656:
649:
645:
638:
634:
630:
626:
619:
615:
611:
604:
599:
597:
593:
586:
582:
578:
574:
567:
559:
557:
550:
547:
546:
545:
542:
538:
531:
523:
521:
519:
515:
511:
495:
492:
486:
480:
454:
447:
443:
435:
432:
426:
420:
413:
412:
411:
394:
388:
357:
349:
345:
341:
338:
335:
327:
323:
316:
310:
307:
304:
296:
292:
288:
285:
278:
277:
276:
275:
271:
267:
247:
239:
231:
228:
225:
217:
214:
206:
201:
197:
191:
188:
183:
177:
171:
164:
163:
162:
160:
156:
152:
148:
144:
140:
133:
128:
126:
122:
118:
114:
110:
106:
102:
98:
94:
90:
85:
81:
74:
71:
67:
59:
57:
55:
51:
47:
43:
39:
38:Roger A. Horn
35:
31:
27:
26:prime numbers
23:
19:
18:number theory
1204:Bateman–Horn
1203:
1115:
1090:
1086:
1056:
1011:
1007:
996:is sampled.
993:
989:
985:
981:
977:
973:
969:
965:
958:
954:
947:
945:
899:
895:
891:
890:elements of
887:
883:
879:
875:
870:
866:
862:
858:
856:
666:
654:
647:
643:
636:
632:
628:
624:
617:
613:
609:
602:
600:
591:
584:
580:
576:
572:
565:
563:
554:
540:
536:
529:
527:
517:
513:
509:
472:
380:
273:
269:
265:
263:
158:
154:
150:
146:
142:
135:
131:
129:
124:
120:
116:
112:
108:
104:
100:
96:
92:
83:
79:
72:
65:
63:
49:
21:
15:
1298:Oppermann's
1244:Gilbreath's
1214:Bunyakovsky
992:over which
659:twin primes
130:An integer
46:twin primes
30:polynomials
1333:Categories
1303:Polignac's
1276:Twin prime
1271:Legendre's
1259:Goldbach's
1189:Agoh–Giuga
1125:1807.08899
1079:1058.11001
1046:0105.03302
1000:References
583:for which
60:Definition
1288:Lemoine's
1229:Dickson's
1209:Brocard's
1194:Andrica's
818:
800:≈
781:
750:−
733:−
713:≥
706:∏
699:∼
681:π
433:≡
339:−
308:−
293:∏
229:
198:∫
184:∼
68:distinct
1293:Mersenne
1224:Cramér's
1055:(2004),
560:Examples
1249:Grimm's
1199:Artin's
1095:Bibcode
1038:0148632
1030:2004056
1077:
1067:
1044:
1036:
1028:
646:) and
458:
366:
264:where
20:, the
1308:Pólya
1120:arXiv
1026:JSTOR
878:) in
381:with
1264:weak
1065:ISBN
803:1.32
616:and
493:<
36:and
1182:2nd
1177:1st
1103:doi
1091:435
1075:Zbl
1042:Zbl
1016:doi
946:in
815:log
778:log
669:is
444:mod
226:log
16:In
1335::
1101:,
1089:,
1073:,
1063:,
1040:,
1034:MR
1032:,
1024:,
1012:16
1010:,
970:no
598:.
127:.
56:.
1155:e
1148:t
1141:v
1122::
1110:.
1105::
1097::
1018::
994:x
990:F
986:d
982:d
978:F
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966:x
962:3
959:F
955:x
951:3
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928:u
925:+
920:3
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896:F
892:F
888:x
884:F
880:F
876:x
874:(
871:i
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837:.
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821:x
812:(
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