214:" does not mean that there exists any infinitely long arithmetic progression of prime numbers (there is not), nor that there exists any particular arithmetic progression of prime numbers that is in some sense "arbitrarily long". Rather, the phrase is used to refer to the fact that no matter how large a number
704:
605:
384:
620:
32:
are used in statements to make clear the fact that an object is large, small, or long with little limitation or restraint, respectively. The use of "arbitrarily" often occurs in the context of
521:
713:". For example, although prime numbers can be arbitrarily large, an infinitely large prime number does not exist—since all prime numbers (as well as all other integers) are finite.
296:
871:
794:
743:
491:
442:
288:
160:
79:
837:
815:
764:
512:
412:
255:
233:
203:
181:
130:
100:
210:
In the common parlance, the term "arbitrarily long" is often used in the context of sequence of numbers. For example, to say that there are "arbitrarily long
840:
is." In these cases, the phrase "arbitrarily large" does not have the meaning indicated above (i.e., "however large a number, there will be
908:
457:". For instance, while it is true that prime numbers can be arbitrarily large (since there are infinitely many of them due to
699:{\displaystyle \exists n\in \mathbb {R} {\mbox{ such that }}\forall x\in \mathbb {R} {\mbox{, }}x>n\Rightarrow f(x)\geq 0}
600:{\displaystyle \forall n\in \mathbb {R} {\mbox{, }}\exists x\in \mathbb {R} {\mbox{ such that }}x>n\land f(x)\geq 0}
978:
211:
379:{\displaystyle \forall \epsilon \in \mathbb {R} _{+},\,\exists x\in \mathbb {R} :|x|<\epsilon \land P(x)}
905:
873:
still holds."). Instead, the usage in this case is in fact logically synonymous with "all".
847:
770:
719:
467:
418:
264:
136:
55:
8:
887:
458:
882:
822:
800:
749:
611:
497:
454:
397:
240:
218:
188:
166:
115:
85:
40:
thereof), though its meaning can differ from that of "sufficiently" and "infinitely".
949:
925:
710:
912:
236:
is, there exists some arithmetic progression of prime numbers of length at least
448:
972:
33:
17:
461:), it is not true that all sufficiently large numbers are prime.
37:
261:
Similar to arbitrarily large, one can also define the phrase "
449:
Arbitrarily large vs. sufficiently large vs. infinitely large
453:
While similar, "arbitrarily large" is not equivalent to "
290:
holds for arbitrarily small real numbers", as follows:
660:
639:
561:
540:
850:
825:
803:
773:
752:
722:
709:
Furthermore, "arbitrarily large" also does not mean "
623:
524:
500:
470:
421:
400:
299:
267:
243:
221:
191:
169:
139:
118:
88:
58:
865:
831:
809:
788:
758:
737:
698:
599:
506:
485:
436:
406:
378:
282:
249:
227:
197:
175:
154:
124:
94:
73:
970:
716:In some cases, phrases such as "the proposition
393:However small a number, there will be a number
950:"Definition:Arbitrarily Large - ProofWiki"
926:"Definition:Arbitrarily Small - ProofWiki"
767:" are used primarily for emphasis, as in "
655:
634:
556:
535:
335:
324:
311:
212:arithmetic progressions of prime numbers
971:
493:is non-negative for arbitrarily large
81:is non-negative for arbitrarily large
464:As another example, the statement "
13:
645:
624:
546:
525:
325:
300:
162:is non-negative for some value of
14:
990:
942:
918:
899:
860:
854:
783:
777:
745:is true for arbitrarily large
732:
726:
687:
681:
675:
588:
582:
480:
474:
431:
425:
373:
367:
351:
343:
277:
271:
149:
143:
68:
62:
1:
893:
614:", the same phrase becomes:
7:
876:
43:
10:
995:
911:February 22, 2012, at the
515:." could be rewritten as:
415:smaller than it such that
915:Accessed 21 February 2012
906:4 Arbitrarily Large Data.
979:Mathematical terminology
844:larger number for which
111:"For every real number
867:
833:
818:, no matter how large
811:
790:
760:
739:
700:
601:
508:
487:
438:
408:
380:
284:
251:
229:
199:
177:
156:
126:
96:
75:
868:
834:
812:
791:
761:
740:
701:
641: such that
602:
563: such that
509:
488:
439:
409:
381:
285:
252:
230:
200:
178:
157:
127:
97:
76:
866:{\displaystyle P(x)}
848:
823:
801:
789:{\displaystyle P(x)}
771:
750:
738:{\displaystyle P(x)}
720:
621:
522:
498:
486:{\displaystyle f(x)}
468:
437:{\displaystyle P(x)}
419:
398:
297:
283:{\displaystyle P(x)}
265:
241:
219:
189:
167:
155:{\displaystyle f(x)}
137:
116:
107:is a shorthand for:
86:
74:{\displaystyle f(x)}
56:
888:Mathematical jargon
883:Sufficiently large
863:
829:
807:
786:
756:
735:
696:
664:
643:
612:sufficiently large
597:
565:
544:
504:
483:
455:sufficiently large
434:
404:
376:
280:
247:
225:
195:
173:
152:
122:
92:
71:
832:{\displaystyle x}
810:{\displaystyle x}
759:{\displaystyle x}
663:
642:
564:
543:
507:{\displaystyle x}
407:{\displaystyle x}
250:{\displaystyle n}
228:{\displaystyle n}
198:{\displaystyle n}
176:{\displaystyle x}
125:{\displaystyle n}
95:{\displaystyle x}
26:arbitrarily small
22:arbitrarily large
986:
964:
963:
961:
960:
946:
940:
939:
937:
936:
922:
916:
903:
872:
870:
869:
864:
838:
836:
835:
830:
816:
814:
813:
808:
796:is true for all
795:
793:
792:
787:
765:
763:
762:
757:
744:
742:
741:
736:
711:infinitely large
705:
703:
702:
697:
665:
661:
658:
644:
640:
637:
610:However, using "
606:
604:
603:
598:
566:
562:
559:
545:
541:
538:
513:
511:
510:
505:
492:
490:
489:
484:
459:Euclid's theorem
443:
441:
440:
435:
413:
411:
410:
405:
389:In other words:
385:
383:
382:
377:
354:
346:
338:
320:
319:
314:
289:
287:
286:
281:
256:
254:
253:
248:
234:
232:
231:
226:
204:
202:
201:
196:
182:
180:
179:
174:
161:
159:
158:
153:
131:
129:
128:
123:
101:
99:
98:
93:
80:
78:
77:
72:
30:arbitrarily long
994:
993:
989:
988:
987:
985:
984:
983:
969:
968:
967:
958:
956:
948:
947:
943:
934:
932:
924:
923:
919:
913:Wayback Machine
904:
900:
896:
879:
849:
846:
845:
824:
821:
820:
802:
799:
798:
772:
769:
768:
751:
748:
747:
721:
718:
717:
659:
654:
638:
633:
622:
619:
618:
560:
555:
539:
534:
523:
520:
519:
499:
496:
495:
469:
466:
465:
451:
420:
417:
416:
399:
396:
395:
350:
342:
334:
315:
310:
309:
298:
295:
294:
266:
263:
262:
242:
239:
238:
220:
217:
216:
190:
187:
186:
168:
165:
164:
138:
135:
134:
117:
114:
113:
87:
84:
83:
57:
54:
53:
46:
12:
11:
5:
992:
982:
981:
966:
965:
941:
917:
897:
895:
892:
891:
890:
885:
878:
875:
862:
859:
856:
853:
828:
806:
785:
782:
779:
776:
755:
734:
731:
728:
725:
707:
706:
695:
692:
689:
686:
683:
680:
677:
674:
671:
668:
657:
653:
650:
647:
636:
632:
629:
626:
608:
607:
596:
593:
590:
587:
584:
581:
578:
575:
572:
569:
558:
554:
551:
548:
537:
533:
530:
527:
503:
482:
479:
476:
473:
450:
447:
446:
445:
433:
430:
427:
424:
403:
387:
386:
375:
372:
369:
366:
363:
360:
357:
353:
349:
345:
341:
337:
333:
330:
327:
323:
318:
313:
308:
305:
302:
279:
276:
273:
270:
246:
224:
208:
207:
194:
172:
151:
148:
145:
142:
121:
105:
104:
91:
70:
67:
64:
61:
48:The statement
45:
42:
20:, the phrases
9:
6:
4:
3:
2:
991:
980:
977:
976:
974:
955:
954:proofwiki.org
951:
945:
931:
930:proofwiki.org
927:
921:
914:
910:
907:
902:
898:
889:
886:
884:
881:
880:
874:
857:
851:
843:
839:
826:
817:
804:
780:
774:
766:
753:
729:
723:
714:
712:
693:
690:
684:
678:
672:
669:
666:
651:
648:
630:
627:
617:
616:
615:
613:
594:
591:
585:
579:
576:
573:
570:
567:
552:
549:
531:
528:
518:
517:
516:
514:
501:
477:
471:
462:
460:
456:
428:
422:
414:
401:
392:
391:
390:
370:
364:
361:
358:
355:
347:
339:
331:
328:
321:
316:
306:
303:
293:
292:
291:
274:
268:
259:
257:
244:
235:
222:
213:
205:
192:
184:greater than
183:
170:
146:
140:
132:
119:
110:
109:
108:
102:
89:
65:
59:
51:
50:
49:
41:
39:
35:
31:
27:
23:
19:
957:. Retrieved
953:
944:
933:. Retrieved
929:
920:
901:
841:
819:
797:
746:
715:
708:
609:
494:
463:
452:
394:
388:
260:
237:
215:
209:
185:
163:
112:
106:
82:
47:
34:real numbers
29:
25:
21:
15:
18:mathematics
959:2019-11-19
935:2019-11-19
894:References
691:≥
676:⇒
652:∈
646:∀
631:∈
625:∃
592:≥
577:∧
553:∈
547:∃
532:∈
526:∀
362:∧
359:ϵ
332:∈
326:∃
307:∈
304:ϵ
301:∀
36:(and its
973:Category
909:Archived
877:See also
44:Examples
662:,
542:,
38:subsets
444:holds.
842:some
670:>
571:>
356:<
28:and
16:In
975::
952:.
928:.
258:.
206:."
133:,
103:."
24:,
962:.
938:.
861:)
858:x
855:(
852:P
827:x
805:x
784:)
781:x
778:(
775:P
754:x
733:)
730:x
727:(
724:P
694:0
688:)
685:x
682:(
679:f
673:n
667:x
656:R
649:x
635:R
628:n
595:0
589:)
586:x
583:(
580:f
574:n
568:x
557:R
550:x
536:R
529:n
502:x
481:)
478:x
475:(
472:f
432:)
429:x
426:(
423:P
402:x
374:)
371:x
368:(
365:P
352:|
348:x
344:|
340::
336:R
329:x
322:,
317:+
312:R
278:)
275:x
272:(
269:P
245:n
223:n
193:n
171:x
150:)
147:x
144:(
141:f
120:n
90:x
69:)
66:x
63:(
60:f
52:"
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.