213.143.133.108 at 09:57, 20 June 2013

2013-06-20T09:57:07Z

← Older revision		Revision as of 11:57, 20 June 2013
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	Hi there, ~~I am Felicidad Oquendo~~. ~~One~~ of ~~my favorite hobbies~~ is ~~tenting~~ and ~~now I~~'~~m attempting to make money with it~~. ~~Some time~~ in the ~~past he chose~~ to ~~live in Kansas~~. ~~After becoming out~~ of ~~my job for years I grew~~ to ~~become a production~~ and ~~distribution officer~~ but ~~I plan on changing it~~.<br><br>~~Visit my web page~~ ... [~~https~~://~~Deathcontrol~~.de/~~index/users/view/id/2759 extended car warranty~~]		The '''Euclid–Mullin sequence''' is an infinite sequence of distinct [[prime number]]s, in which each element is the least [[prime factor]] of one plus the product of all earlier elements. They are named after the ancient Greek mathematician [[Euclid]], because their definition relies on an idea in [[Euclid's theorem\|Euclid's proof that there are infinitely many primes]], and after [[Albert A. Mullin]], who asked about the sequence in 1963.<ref>{{citation\|first=Albert A.\|last=Mullin\|contribution=Recursive function theory (A modern look at a Euclidean idea)\|title=Research problems\|journal=[[Bulletin of the American Mathematical Society]]\|volume=69\|issue=6\|year=1963\|page=737\|doi=10.1090/S0002-9904-1963-11017-4 }}.</ref>

			The first 51 elements of the sequence are

			: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, 87991098722552272708281251793312351581099392851768893748012603709343,<ref>[http://www.mersenneforum.org/showthread.php?p=207854#post207854 Factoring 43rd term of Euclid–Mullin sequence]</ref><ref>http://escatter11.fullerton.edu/nfs/forum_thread.php?id=102&postid=398#398</ref> 107, 127, 3313, 227432689108589532754984915075774848386671439568260420754414940780761245893,<ref name=em47>[http://www.mersenneforum.org/showpost.php?p=311145&postcount=52 Factoring EM47]</ref> 59, 31, 211... {{OEIS\|id=A000945}}

			These are the only known elements {{As of\|2012\|9\|lc=on}}. Finding the next one requires finding the least prime factor of a 335-digit number (which is known to be [[composite number\|composite]]).

			==Definition==
			If ''a<sub>n</sub>'' denotes the ''n''-th element of the sequence, then ''a<sub>n</sub>'' is the least prime factor of

			:<math>\left(\prod_{i < n} a_i\right)+1\,.</math>

			The first element is therefore the least prime factor of the [[empty product]] plus one, which is 2. The element 13 in the sequence is the least prime factor of 2 × 3 × 7 × 43 + 1 = 1806 + 1 = 1807 = 13 × 139.

			==Properties==
			The sequence is infinitely long and does not contain repeated elements. This can be proved using the method of [[Euclid]]'s proof that [[Euclid's theorem\|there are infinitely many primes]]. That proof is [[Constructive proof\|constructive]], and the sequence is the result of performing a version of that construction.

			==Conjecture==
			{{unsolved\|mathematics\|Does every prime number appear in the Euclid–Mullin sequence?}}
			{{harvtxt\|Mullin\|1963}} 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 [[Computable function\|computable]]; these problems both remain open. The least prime number not known to be an element of the sequence is 41.

			The positions of the prime numbers from 2 to 97 are:
			: 2:1, 3:2, 5:7, 7:3, 11:12, 13:5, 17:13, 19:36, 23:25, 29:33, 31:50,<ref name=em47 /> 37:18, 41:?, 43:4, 47:?, 53:6, 59:49,<ref name=em47 /> 61:42, 67:?, 71:22, 73:?, 79:?, 83:?, 89:35, 97:26 {{OEIS\|id=A056756}}
			where ? indicates that the position (or whether it exists at all) is unknown as of 2012.<ref>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.</ref>

			==Related sequences==
			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 monotonic.<ref>{{citation
			\| last = Naur \| first = Thorkil
			\| doi = 10.2307/2044665
			\| issue = 1
			\| journal = Proceedings of the American Mathematical Society
			\| mr = 722412
			\| pages = 43–44
			\| title = Mullin's sequence of primes is not monotonic
			\| volume = 90
			\| year = 1984}}.</ref> The numbers in this sequence are
			:2, 3, 7, 43, 139, 50207, 340999, 2365347734339, 4680225641471129, 1368845206580129, 889340324577880670089824574922371, … {{OEIS\|id=A000946}}.

			Alternatively, taking each number to be one plus the product of the previous numbers (rather than factoring it) gives [[Sylvester's sequence]]. 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.<ref>{{citation
			\| last1 = Guy \| first1 = Richard
			\| last2 = Nowakowski \| first2 = Richard
			\| issue = 2
			\| journal = Delta (Waukesha)
			\| mr = 0384675
			\| pages = 49–63
			\| title = Discovering primes with Euclid
			\| volume = 5
			\| year = 1975}}.</ref>

			==See also==
			*[[Euclid number]]

			==References==
			{{reflist}}

			==External links==
			*{{MathWorld \|title=Euclid–Mullin Sequence \|urlname=Euclid-MullinSequence}}
			*[http://mersenneforum.org/showpost.php?p=60960&postcount=65 ''Factoring 43rd Term of Euclid–Mullin sequence''], factorizations of the numbers for which the sequence elements is the smallest prime factor

			{{DEFAULTSORT:Euclid-Mullin sequence}}
			[[Category:Integer sequences]]
			[[Category:Prime numbers]]
			[[Category:Unsolved problems in mathematics]]

137.205.222.193 at 17:27, 27 February 2011

2011-02-27T17:27:48Z

New page

Hi there, I am Felicidad Oquendo. One of my favorite hobbies is tenting and now I'm attempting to make money with it. Some time in the past he chose to live in Kansas. After becoming out of my job for years I grew to become a production and distribution officer but I plan on changing it.<br><br>Visit my web page ... [https://Deathcontrol.de/index/users/view/id/2759 extended car warranty]

P-wave modulus - Revision history

213.143.133.108 at 09:57, 20 June 2013

137.205.222.193 at 17:27, 27 February 2011