270:
733:
147:
655:
516:
399:
51:
587:
457:
426:
340:
837:
607:
560:
536:
299:
39:
674:
880:
657:
satisfies these conditions, so the constant 0.003712192634435363748597110122136... is normal in base 10, and 0.003101525354661104...
940:
265:{\displaystyle \displaystyle \sum _{n=1}^{\infty }p_{n}10^{-\left(n+\sum _{k=1}^{n}\lfloor \log _{10}{p_{k}}\rfloor \right)}}
765:
930:
612:
828:
538:
is any natural number greater than or equal to 2, then the constant obtained by concatenating "0." with the
131:
55:
935:
925:
73:
771:
466:
349:
63:
294:
898:
876:
59:
47:
901:
846:
565:
435:
404:
318:
791:
Copeland and Erdős considered 1 a prime, and they defined the constant as 0.12357111317...
72:
By a similar argument, any constant created by concatenating "0." with all primes in an
69:
is a sum of at most six primes. It also follows directly from its normality (see below).
592:
545:
521:
460:
315:
Copeland and Erdős's proof that their constant is normal relies only on the fact that
919:
869:
832:
135:
127:
851:
864:
287:
123:, so the concatenated primes contain arbitrarily long sequences of the digit zero.
25:
768:: the truncated value of this constant multiplied by the appropriate power of 10.
860:
539:
906:
28:
in order. Its value, using the modern definition of prime, is approximately
343:
88:
66:
21:
103: + 1. By Dirichlet's theorem, the arithmetic progression
728:{\displaystyle \displaystyle \sum _{n=1}^{\infty }b^{-p_{n}},\,}
95:
and to 10, will be irrational; for example, primes of the form 4
303:
34:
774:: concatenating all natural numbers, not just primes.
678:
677:
615:
595:
568:
548:
524:
469:
438:
407:
352:
321:
151:
150:
868:
727:
649:
601:
581:
554:
530:
510:
451:
420:
393:
334:
264:
52:Dirichlet's theorem on arithmetic progressions
838:Bulletin of the American Mathematical Society
800:
650:{\displaystyle \lfloor n(\log n)^{2}\rfloor }
8:
644:
616:
251:
223:
811:
138:in 1946 (hence the name of the constant).
875:(5th ed.), Oxford University Press,
850:
723:
712:
704:
694:
683:
676:
638:
614:
594:
573:
567:
547:
523:
487:
474:
468:
443:
437:
412:
406:
370:
357:
351:
326:
320:
244:
239:
230:
217:
206:
187:
177:
167:
156:
149:
871:An Introduction to the Theory of Numbers
32:0.235711131719232931374143... (sequence
784:
459:is any strictly increasing sequence of
20:is the concatenation of "0." with the
7:
835:(1946), "Note on Normal Numbers",
695:
168:
14:
432:prime number. More generally, if
511:{\displaystyle s_{n}=n^{1+o(1)}}
394:{\displaystyle p_{n}=n^{1+o(1)}}
852:10.1090/S0002-9904-1946-08657-7
115:, and those primes are also in
635:
622:
503:
497:
386:
380:
126:In base 10, the constant is a
58:(Hardy and Wright, p. 113) or
1:
752:th digit is 1 if and only if
738:which can be written in base
609:. For example, the sequence
742:as 0.0110101000101000101...
957:
766:Smarandache–Wellin numbers
50:; this can be proven with
902:"Copeland-Erdos Constant"
801:Copeland & Erdős 1946
756:is prime, is irrational.
141:The constant is given by
111:contains primes for all
812:Hardy & Wright 1979
562:representations of the
132:Arthur Herbert Copeland
107: · 10 +
24:representations of the
18:Copeland–Erdős constant
941:Mathematical constants
729:
699:
651:
603:
583:
556:
532:
512:
453:
422:
395:
336:
266:
222:
172:
74:arithmetic progression
772:Champernowne constant
730:
679:
661:is normal in base 7.
652:
604:
589:'s is normal in base
584:
582:{\displaystyle s_{n}}
557:
533:
513:
454:
452:{\displaystyle s_{n}}
423:
421:{\displaystyle p_{n}}
396:
337:
335:{\displaystyle p_{n}}
267:
202:
152:
675:
613:
593:
566:
546:
522:
467:
436:
405:
350:
319:
148:
56:Bertrand's postulate
344:strictly increasing
130:, a fact proven by
99: + 1 or 8
931:Irrational numbers
899:Weisstein, Eric W.
725:
724:
664:In any given base
647:
599:
579:
552:
528:
508:
449:
418:
391:
332:
295:continued fraction
262:
261:
602:{\displaystyle b}
555:{\displaystyle b}
531:{\displaystyle b}
311:Related constants
948:
912:
911:
885:
874:
855:
854:
815:
809:
803:
798:
792:
789:
734:
732:
731:
726:
719:
718:
717:
716:
698:
693:
656:
654:
653:
648:
643:
642:
608:
606:
605:
600:
588:
586:
585:
580:
578:
577:
561:
559:
558:
553:
537:
535:
534:
529:
517:
515:
514:
509:
507:
506:
479:
478:
458:
456:
455:
450:
448:
447:
427:
425:
424:
419:
417:
416:
400:
398:
397:
392:
390:
389:
362:
361:
341:
339:
338:
333:
331:
330:
306:
271:
269:
268:
263:
260:
259:
258:
254:
250:
249:
248:
235:
234:
221:
216:
182:
181:
171:
166:
60:Ramare's theorem
46:The constant is
37:
956:
955:
951:
950:
949:
947:
946:
945:
916:
915:
897:
896:
893:
883:
859:
845:(10): 857–860,
829:Copeland, A. H.
827:
824:
819:
818:
810:
806:
799:
795:
790:
786:
781:
762:
747:
708:
700:
673:
672:
660:
634:
611:
610:
591:
590:
569:
564:
563:
544:
543:
520:
519:
483:
470:
465:
464:
461:natural numbers
439:
434:
433:
408:
403:
402:
366:
353:
348:
347:
322:
317:
316:
313:
298:
280:
240:
226:
195:
191:
183:
173:
146:
145:
33:
12:
11:
5:
954:
952:
944:
943:
938:
933:
928:
918:
917:
914:
913:
892:
891:External links
889:
888:
887:
881:
857:
823:
820:
817:
816:
804:
793:
783:
782:
780:
777:
776:
775:
769:
761:
758:
743:
736:
735:
722:
715:
711:
707:
703:
697:
692:
689:
686:
682:
658:
646:
641:
637:
633:
630:
627:
624:
621:
618:
598:
576:
572:
551:
527:
505:
502:
499:
496:
493:
490:
486:
482:
477:
473:
446:
442:
415:
411:
388:
385:
382:
379:
376:
373:
369:
365:
360:
356:
329:
325:
312:
309:
278:
273:
272:
257:
253:
247:
243:
238:
233:
229:
225:
220:
215:
212:
209:
205:
201:
198:
194:
190:
186:
180:
176:
170:
165:
162:
159:
155:
44:
43:
13:
10:
9:
6:
4:
3:
2:
953:
942:
939:
937:
936:Prime numbers
934:
932:
929:
927:
924:
923:
921:
909:
908:
903:
900:
895:
894:
890:
884:
882:0-19-853171-0
878:
873:
872:
866:
865:Wright, E. M.
862:
858:
853:
848:
844:
840:
839:
834:
830:
826:
825:
821:
814:, p. 112
813:
808:
805:
802:
797:
794:
788:
785:
778:
773:
770:
767:
764:
763:
759:
757:
755:
751:
746:
741:
720:
713:
709:
705:
701:
690:
687:
684:
680:
671:
670:
669:
667:
662:
639:
631:
628:
625:
619:
596:
574:
570:
549:
541:
525:
500:
494:
491:
488:
484:
480:
475:
471:
462:
444:
440:
431:
413:
409:
383:
377:
374:
371:
367:
363:
358:
354:
345:
327:
323:
310:
308:
305:
301:
296:
291:
289:
285:
281:
255:
245:
241:
236:
231:
227:
218:
213:
210:
207:
203:
199:
196:
192:
188:
184:
178:
174:
163:
160:
157:
153:
144:
143:
142:
139:
137:
133:
129:
128:normal number
124:
122:
119: +
118:
114:
110:
106:
102:
98:
94:
90:
86:
82:
79: +
78:
75:
70:
68:
65:
61:
57:
53:
49:
41:
36:
31:
30:
29:
27:
26:prime numbers
23:
19:
905:
870:
861:Hardy, G. H.
842:
836:
807:
796:
787:
753:
749:
744:
739:
737:
665:
663:
429:
314:
292:
288:prime number
283:
276:
274:
140:
125:
120:
116:
112:
108:
104:
100:
96:
92:
84:
80:
76:
71:
45:
17:
15:
668:the number
62:that every
926:Paul Erdős
920:Categories
779:References
748:where the
463:such that
136:Paul Erdős
48:irrational
907:MathWorld
867:(1979) ,
833:Erdős, P.
706:−
696:∞
681:∑
645:⌋
629:
617:⌊
252:⌋
237:
224:⌊
204:∑
189:−
169:∞
154:∑
760:See also
401:, where
83:, where
822:Sources
428:is the
304:A030168
302::
282:is the
89:coprime
67:integer
38:in the
35:A033308
22:base 10
879:
275:where
297:is (
877:ISBN
540:base
518:and
346:and
300:OEIS
293:Its
134:and
64:even
40:OEIS
16:The
847:doi
626:log
342:is
307:).
286:th
228:log
91:to
87:is
54:or
922::
904:.
863:;
843:52
841:,
831:;
290:.
232:10
185:10
117:cd
105:dn
77:dn
42:).
910:.
886:.
856:.
849::
754:n
750:n
745:b
740:b
721:,
714:n
710:p
702:b
691:1
688:=
685:n
666:b
659:7
640:2
636:)
632:n
623:(
620:n
597:b
575:n
571:s
550:b
542:-
526:b
504:)
501:1
498:(
495:o
492:+
489:1
485:n
481:=
476:n
472:s
445:n
441:s
430:n
414:n
410:p
387:)
384:1
381:(
378:o
375:+
372:1
368:n
364:=
359:n
355:p
328:n
324:p
284:n
279:n
277:p
256:)
246:k
242:p
219:n
214:1
211:=
208:k
200:+
197:n
193:(
179:n
175:p
164:1
161:=
158:n
121:a
113:m
109:a
101:n
97:n
93:d
85:a
81:a
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