281:
744:
158:
666:
527:
410:
62:
598:
468:
437:
351:
848:
618:
571:
547:
310:
50:
685:
891:
668:
satisfies these conditions, so the constant 0.003712192634435363748597110122136... is normal in base 10, and 0.003101525354661104...
951:
276:{\displaystyle \displaystyle \sum _{n=1}^{\infty }p_{n}10^{-\left(n+\sum _{k=1}^{n}\lfloor \log _{10}{p_{k}}\rfloor \right)}}
776:
941:
623:
839:
549:
is any natural number greater than or equal to 2, then the constant obtained by concatenating "0." with the
142:
66:
946:
936:
84:
782:
477:
360:
74:
305:
909:
887:
70:
58:
912:
857:
576:
446:
415:
329:
802:
Copeland and Erdős considered 1 a prime, and they defined the constant as 0.12357111317...
83:
By a similar argument, any constant created by concatenating "0." with all primes in an
80:
is a sum of at most six primes. It also follows directly from its normality (see below).
603:
556:
532:
471:
17:
326:
Copeland and Erdős's proof that their constant is normal relies only on the fact that
930:
880:
843:
146:
138:
862:
875:
298:
134:, so the concatenated primes contain arbitrarily long sequences of the digit zero.
36:
779:: the truncated value of this constant multiplied by the appropriate power of 10.
871:
550:
917:
39:
in order. Its value, using the modern definition of prime, is approximately
354:
99:
77:
32:
114: + 1. By Dirichlet's theorem, the arithmetic progression
739:{\displaystyle \displaystyle \sum _{n=1}^{\infty }b^{-p_{n}},\,}
106:
and to 10, will be irrational; for example, primes of the form 4
314:
45:
785:: concatenating all natural numbers, not just primes.
689:
688:
626:
606:
579:
559:
535:
480:
449:
418:
363:
332:
162:
161:
879:
738:
660:
612:
592:
565:
541:
521:
462:
431:
404:
345:
275:
63:Dirichlet's theorem on arithmetic progressions
849:Bulletin of the American Mathematical Society
811:
661:{\displaystyle \lfloor n(\log n)^{2}\rfloor }
8:
655:
627:
262:
234:
822:
149:in 1946 (hence the name of the constant).
886:(5th ed.), Oxford University Press,
861:
734:
723:
715:
705:
694:
687:
649:
625:
605:
584:
578:
558:
534:
498:
485:
479:
454:
448:
423:
417:
381:
368:
362:
337:
331:
255:
250:
241:
228:
217:
198:
188:
178:
167:
160:
882:An Introduction to the Theory of Numbers
43:0.235711131719232931374143... (sequence
795:
470:is any strictly increasing sequence of
31:is the concatenation of "0." with the
7:
846:(1946), "Note on Normal Numbers",
706:
179:
25:
443:prime number. More generally, if
522:{\displaystyle s_{n}=n^{1+o(1)}}
405:{\displaystyle p_{n}=n^{1+o(1)}}
863:10.1090/S0002-9904-1946-08657-7
126:, and those primes are also in
646:
633:
514:
508:
397:
391:
137:In base 10, the constant is a
69:(Hardy and Wright, p. 113) or
1:
763:th digit is 1 if and only if
749:which can be written in base
620:. For example, the sequence
753:as 0.0110101000101000101...
968:
777:Smarandache–Wellin numbers
61:; this can be proven with
913:"Copeland-Erdos Constant"
812:Copeland & Erdős 1946
767:is prime, is irrational.
152:The constant is given by
122:contains primes for all
823:Hardy & Wright 1979
573:representations of the
143:Arthur Herbert Copeland
118: · 10 +
35:representations of the
29:Copeland–Erdős constant
18:Copeland-Erdős constant
952:Mathematical constants
740:
710:
662:
614:
594:
567:
543:
523:
464:
433:
406:
347:
277:
233:
183:
85:arithmetic progression
783:Champernowne constant
741:
690:
672:is normal in base 7.
663:
615:
600:'s is normal in base
595:
593:{\displaystyle s_{n}}
568:
544:
524:
465:
463:{\displaystyle s_{n}}
434:
432:{\displaystyle p_{n}}
407:
348:
346:{\displaystyle p_{n}}
278:
213:
163:
686:
624:
604:
577:
557:
533:
478:
447:
416:
361:
330:
159:
67:Bertrand's postulate
355:strictly increasing
141:, a fact proven by
110: + 1 or 8
942:Irrational numbers
910:Weisstein, Eric W.
736:
735:
675:In any given base
658:
610:
590:
563:
539:
519:
460:
429:
402:
343:
306:continued fraction
273:
272:
613:{\displaystyle b}
566:{\displaystyle b}
542:{\displaystyle b}
322:Related constants
16:(Redirected from
959:
923:
922:
896:
885:
866:
865:
826:
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814:
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803:
800:
745:
743:
742:
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728:
727:
709:
704:
667:
665:
664:
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619:
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611:
599:
597:
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588:
572:
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517:
490:
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317:
282:
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261:
260:
259:
246:
245:
232:
227:
193:
192:
182:
177:
71:Ramare's theorem
57:The constant is
48:
21:
967:
966:
962:
961:
960:
958:
957:
956:
927:
926:
908:
907:
904:
894:
870:
856:(10): 857–860,
840:Copeland, A. H.
838:
835:
830:
829:
821:
817:
810:
806:
801:
797:
792:
773:
758:
719:
711:
684:
683:
671:
645:
622:
621:
602:
601:
580:
575:
574:
555:
554:
531:
530:
494:
481:
476:
475:
472:natural numbers
450:
445:
444:
419:
414:
413:
377:
364:
359:
358:
333:
328:
327:
324:
309:
291:
251:
237:
206:
202:
194:
184:
157:
156:
44:
23:
22:
15:
12:
11:
5:
965:
963:
955:
954:
949:
944:
939:
929:
928:
925:
924:
903:
902:External links
900:
899:
898:
892:
868:
834:
831:
828:
827:
815:
804:
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793:
791:
788:
787:
786:
780:
772:
769:
754:
747:
746:
733:
726:
722:
718:
714:
708:
703:
700:
697:
693:
669:
657:
652:
648:
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641:
638:
635:
632:
629:
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583:
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488:
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457:
453:
426:
422:
399:
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371:
367:
340:
336:
323:
320:
289:
284:
283:
268:
264:
258:
254:
249:
244:
240:
236:
231:
226:
223:
220:
216:
212:
209:
205:
201:
197:
191:
187:
181:
176:
173:
170:
166:
55:
54:
24:
14:
13:
10:
9:
6:
4:
3:
2:
964:
953:
950:
948:
947:Prime numbers
945:
943:
940:
938:
935:
934:
932:
920:
919:
914:
911:
906:
905:
901:
895:
893:0-19-853171-0
889:
884:
883:
877:
876:Wright, E. M.
873:
869:
864:
859:
855:
851:
850:
845:
841:
837:
836:
832:
825:, p. 112
824:
819:
816:
813:
808:
805:
799:
796:
789:
784:
781:
778:
775:
774:
770:
768:
766:
762:
757:
752:
731:
724:
720:
716:
712:
701:
698:
695:
691:
682:
681:
680:
678:
673:
650:
642:
639:
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630:
607:
585:
581:
560:
552:
536:
511:
505:
502:
499:
495:
491:
486:
482:
473:
455:
451:
442:
424:
420:
394:
388:
385:
382:
378:
374:
369:
365:
356:
338:
334:
321:
319:
316:
312:
307:
302:
300:
296:
292:
266:
256:
252:
247:
242:
238:
229:
224:
221:
218:
214:
210:
207:
203:
199:
195:
189:
185:
174:
171:
168:
164:
155:
154:
153:
150:
148:
144:
140:
139:normal number
135:
133:
130: +
129:
125:
121:
117:
113:
109:
105:
101:
97:
93:
90: +
89:
86:
81:
79:
76:
72:
68:
64:
60:
52:
47:
42:
41:
40:
38:
37:prime numbers
34:
30:
19:
916:
881:
872:Hardy, G. H.
853:
847:
818:
807:
798:
764:
760:
755:
750:
748:
676:
674:
440:
325:
303:
299:prime number
294:
287:
285:
151:
136:
131:
127:
123:
119:
115:
111:
107:
103:
95:
91:
87:
82:
56:
28:
26:
679:the number
73:that every
937:Paul Erdős
931:Categories
790:References
759:where the
474:such that
147:Paul Erdős
59:irrational
918:MathWorld
878:(1979) ,
844:Erdős, P.
717:−
707:∞
692:∑
656:⌋
640:
628:⌊
263:⌋
248:
235:⌊
215:∑
200:−
180:∞
165:∑
771:See also
412:, where
94:, where
833:Sources
439:is the
315:A030168
313::
293:is the
100:coprime
78:integer
49:in the
46:A033308
33:base 10
890:
286:where
308:is (
888:ISBN
551:base
529:and
357:and
311:OEIS
304:Its
145:and
75:even
51:OEIS
27:The
858:doi
637:log
353:is
318:).
297:th
239:log
102:to
98:is
65:or
933::
915:.
874:;
854:52
852:,
842:;
301:.
243:10
196:10
128:cd
116:dn
88:dn
53:).
921:.
897:.
867:.
860::
765:n
761:n
756:b
751:b
732:,
725:n
721:p
713:b
702:1
699:=
696:n
677:b
670:7
651:2
647:)
643:n
634:(
631:n
608:b
586:n
582:s
561:b
553:-
537:b
515:)
512:1
509:(
506:o
503:+
500:1
496:n
492:=
487:n
483:s
456:n
452:s
441:n
425:n
421:p
398:)
395:1
392:(
389:o
386:+
383:1
379:n
375:=
370:n
366:p
339:n
335:p
295:n
290:n
288:p
267:)
257:k
253:p
230:n
225:1
222:=
219:k
211:+
208:n
204:(
190:n
186:p
175:1
172:=
169:n
132:a
124:m
120:a
112:n
108:n
104:d
96:a
92:a
20:)
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