269:, on the basis of heuristic assumptions that the distribution of primes is random, that every prime does appear in the sequence. However, although similar recursive sequences over other domains do not contain all primes, these problems both remain open for the original Euclid–Mullin sequence. The least prime number not known to be an element of the sequence is 41.
206:
of two primes, 13 × 139. Of these two primes, 13 is the smallest and so included in the sequence. Similarly, the seventh element, 5, is the result of (2 × 3 × 7 × 43 × 13 × 53) + 1 = 1244335, the prime factors of which are 5 and 248867. These examples illustrate why the sequence can leap from very large to very small numbers.
350:. The sequence constructed by repeatedly appending all factors of one plus the product of the previous numbers is the same as the sequence of prime factors of Sylvester's sequence. Like the Euclid–Mullin sequence, this is a non-monotonic sequence of primes, but it is known not to include all primes.
205:
plus one, which is 2. The third element is (2 × 3) + 1 = 7. A better illustration is the fifth element in the sequence, 13. This is calculated by (2 × 3 × 7 × 43) + 1 = 1806 + 1 = 1807, the product
54:
2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, 2801, 11, 17, 5471, 52662739, 23003, 30693651606209, 37, 1741, 1313797957, 887, 71, 7127, 109, 23, 97, 159227, 643679794963466223081509857, 103, 1079990819, 9539, 3143065813, 29, 3847, 89, 19, 577, 223, 139703, 457, 9649, 61, 4357,
298:
A related sequence of numbers determined by the largest prime factor of one plus the product of the previous numbers (rather than the smallest prime factor) is also known as the Euclid–Mullin sequence. It grows more quickly, but is not
342:
It is also possible to generate modified versions of the Euclid–Mullin sequence by using the same rule of choosing the smallest prime factor at each step, but beginning with a different prime than 2.
196:
55:
87991098722552272708281251793312351581099392851768893748012603709343, 107, 127, 3313, 227432689108589532754984915075774848386671439568260420754414940780761245893, 59, 31, 211... (sequence
276:
2:1, 3:2, 5:7, 7:3, 11:12, 13:5, 17:13, 19:36, 23:25, 29:33, 31:50, 37:18, 41:?, 43:4, 47:?, 53:6, 59:49, 61:42, 67:?, 71:22, 73:?, 79:?, 83:?, 89:35, 97:26 (sequence
69:
These are the only known elements as of
September 2012. Finding the next one requires finding the least prime factor of a 335-digit number (which is known to be
126:
253:
asked whether every prime number appears in the Euclid–Mullin sequence and, if not, whether the problem of testing a given prime for membership in the sequence is
380:
99:
808:
522:
The listing with the question marks is given in the
Extensions field of the OEIS entry, whereas the main listing stops at 33 and has no question marks.
332:
314:
283:
62:
244:
722:
307:
2, 3, 7, 43, 139, 50207, 340999, 2365347734339, 4680225641471129, 1368845206580129, 889340324577880670089824574922371, … (sequence
134:
214:
The sequence is infinitely long and does not contain repeated elements. This can be proved using the method of Euclid's
346:
Alternatively, taking each number to be one plus the product of the previous numbers (rather than factoring it) gives
798:
378:
Mullin, Albert A. (1963), "Recursive function theory (A modern look at a
Euclidean idea)", Research problems,
347:
803:
576:
35:
of one plus the product of all earlier elements. They are named after the ancient Greek mathematician
254:
219:
40:
693:
650:
624:
549:
495:
468:
223:
215:
771:
718:
290:
where ? indicates that the position (or whether it exists at all) is unknown as of 2012.
712:
677:
634:
588:
541:
486:
Booker, Andrew R. (2016), "A variant of the Euclid-Mullin sequence containing every prime",
452:
389:
70:
44:
24:
775:
753:
689:
646:
602:
561:
509:
464:
425:
104:
749:
685:
642:
598:
557:
505:
460:
421:
681:
737:
84:
792:
654:
409:
359:
321:
Not every prime number appears in this sequence, and the sequence of missing primes,
258:
202:
472:
394:
697:
456:
32:
28:
325:
5, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 61, 67, 71, 73, ... (sequence
440:
638:
226:, and the sequence is the result of performing a version of that construction.
593:
266:
780:
668:
Pollack, Paul; Treviño, Enrique (2014), "The primes that Euclid forgot",
300:
553:
233:
532:
Naur, Thorkil (1984), "Mullin's sequence of primes is not monotonic",
36:
545:
615:
Booker, Andrew R. (2012), "On Mullin's second sequence of primes",
500:
629:
740:; Nowakowski, Richard (1975), "Discovering primes with Euclid",
414:
Bulletin of the
Institute of Combinatorics and Its Applications
241:
Does every prime number appear in the Euclid–Mullin sequence?
201:
The first element is therefore the least prime factor of the
327:
309:
278:
191:{\displaystyle {\Bigl (}\prod _{i<n}a_{i}{\Bigr )}+1\,.}
57:
441:"Euclid prime sequences over unique factorization domains"
272:
The positions of the prime numbers from 2 to 97 are:
137:
107:
87:
41:
Euclid's proof that there are infinitely many primes
190:
120:
93:
173:
140:
534:Proceedings of the American Mathematical Society
39:, because their definition relies on an idea in
581:Journal of the Australian Mathematical Society
439:Kurokawa, Nobushige; Satoh, Takakazu (2008),
381:Bulletin of the American Mathematical Society
8:
717:, Cambridge University Press, p. 26,
579:(1968), "On a sequence of prime numbers",
50:The first 51 elements of the sequence are
628:
592:
499:
393:
184:
172:
171:
165:
149:
139:
138:
136:
112:
106:
86:
47:, who asked about the sequence in 1963.
370:
245:(more unsolved problems in mathematics)
262:
250:
31:, in which each element is the least
7:
682:10.4169/amer.math.monthly.121.05.433
303:. The numbers in this sequence are
809:Unsolved problems in number theory
16:Infinite sequence of prime numbers
14:
339:has been proven to be infinite.
220:there are infinitely many primes
395:10.1090/S0002-9904-1963-11017-4
236:Unsolved problem in mathematics
128:, is the least prime factor of
457:10.1080/10586458.2008.10129035
1:
670:American Mathematical Monthly
488:Journal of Integer Sequences
101:th element of the sequence,
412:(1991), "Euclid's primes",
825:
711:Sheppard, Barnaby (2014),
639:10.1515/integers-2012-0034
594:10.1017/S1446788700006236
776:"Euclid–Mullin Sequence"
494:(6): Article 16.6.4, 6,
445:Experimental Mathematics
577:Van der Poorten, A. J.
192:
122:
95:
21:Euclid–Mullin sequence
714:The Logic of Infinity
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123:
121:{\displaystyle a_{n}}
96:
348:Sylvester's sequence
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772:Weisstein, Eric W.
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91:
799:Integer sequences
294:Related sequences
259:Daniel Shanks
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94:{\displaystyle n}
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742:Delta (Waukesha)
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623:(6): 1167–1177,
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23:is an infinite
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12:
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5:
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723:
703:
676:(5): 433–437,
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804:Prime numbers
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724:9781139952774
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203:empty product
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131:
130:
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113:
109:
88:
76:
74:
72:
64:
59:
53:
52:
51:
48:
46:
42:
38:
34:
30:
29:prime numbers
26:
22:
779:
748:(2): 49–63,
745:
741:
738:Guy, Richard
732:
713:
706:
673:
669:
663:
620:
616:
610:
584:
580:
575:Cox, C. D.;
570:
540:(1): 43–44,
537:
533:
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518:
491:
487:
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444:
434:
417:
413:
404:
385:
379:
373:
345:
341:
338:
320:
297:
289:
271:
249:
224:constructive
213:
200:
80:
68:
49:
43:, and after
33:prime factor
27:of distinct
20:
18:
267:conjectured
793:Categories
501:1605.08929
388:(6): 737,
366:References
255:computable
230:Conjecture
210:Properties
77:Definition
781:MathWorld
655:119144088
630:1107.3318
420:: 33–36,
301:monotonic
147:∏
71:composite
617:Integers
473:12924815
354:See also
25:sequence
754:0384675
698:1335826
690:3193727
647:3011555
603:0228417
562:0722412
554:2044665
510:3546618
465:2433881
426:1103634
331:in the
328:A216227
313:in the
310:A000946
282:in the
279:A056756
261: (
61:in the
58:A000945
752:
721:
696:
688:
653:
645:
601:
560:
552:
508:
471:
463:
424:
37:Euclid
694:S2CID
651:S2CID
625:arXiv
550:JSTOR
496:arXiv
469:S2CID
218:that
216:proof
719:ISBN
333:OEIS
315:OEIS
284:OEIS
263:1991
154:<
81:The
63:OEIS
19:The
678:doi
674:121
635:doi
589:doi
542:doi
453:doi
390:doi
73:).
795::
778:,
774:,
750:MR
744:,
692:,
686:MR
684:,
672:,
649:,
643:MR
641:,
633:,
621:12
619:,
599:MR
597:,
583:,
558:MR
556:,
548:,
538:90
536:,
506:MR
504:,
492:19
490:,
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461:MR
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449:17
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386:69
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257:.
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701:.
680::
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637::
627::
591::
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565:.
544::
513:.
498::
476:.
455::
429:.
418:1
399:.
392::
335:)
286:)
238::
186:.
182:1
179:+
174:)
167:i
163:a
157:n
151:i
141:(
114:n
110:a
89:n
65:)
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