en>Tomruen: Not exactly, doesn't need to be regular Undid revision 641568089 by RobertLovesPi (talk)

2015-01-08T12:25:19Z

Not exactly, doesn't need to be regular Undid revision 641568089 by RobertLovesPi (talk)

Show changes

en>Tomruen: /* Special cases and Generalizations */

2014-03-03T06:33:10Z

Special cases and Generalizations

Show changes

en>Tomruen: /* Special cases and Generalizations */

2014-01-25T06:52:30Z

Special cases and Generalizations

← Older revision		Revision as of 08:52, 25 January 2014
Line 1:		Line 1:
	~~47 year old Swimming Coach~~ or ~~Instructor Giovanni Koehler from Labrador City~~, ~~usually spends time with hobbies~~ and ~~interests~~ for ~~instance guitar~~, ~~como ganhar dinheiro na internet~~ and ~~papercraft~~. ~~Last month just made~~ a ~~journey~~ to ~~Rwenzori Mountains National Park~~.<br><br>~~Also visit my site~~ - [http://~~www~~.~~comoganhardinheiro101~~.~~com ganhando dinheiro na internet~~]		In [[mathematics]], the '''Mersenne conjectures''' concern the characterization of [[prime number]]s of a form called [[Mersenne prime]]s, meaning prime numbers that are a [[power of two]] minus one.

			==Original Mersenne conjecture==
			The original, called '''Mersenne's conjecture''', was a statement by [[Marin Mersenne]] in his ''Cogitata Physica-Mathematica'' (1644; see e.g. Dickson 1919) that the numbers <math>2^n - 1</math> were prime for ''n'' = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257, and were [[composite number\|composite]] for all other positive integers ''n'' < 257. Due to the size of these numbers, Mersenne did not and could not test all of them, nor could his peers in the 17th century. It was eventually determined, after three centuries and the availability of new techniques such as the [[Lucas–Lehmer primality test\|Lucas–Lehmer test]], that Mersenne's conjecture contained five errors, namely two are composite (''n'' = 67, 257) and three omitted primes (''n'' = 61, 89, 107). The correct list is: ''n'' = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107 and 127.

			While Mersenne's original conjecture is false, it has led to the '''New Mersenne conjecture''' and the '''Lenstra–Pomerance–Wagstaff conjecture'''.

			== New Mersenne conjecture ==
			The '''New Mersenne conjecture''' or '''Bateman, Selfridge and Wagstaff conjecture''' (Bateman et al. 1989) states that for any [[Even and odd numbers\|odd]] [[natural number]] ''p'', if any two of the following conditions hold, then so does the third:

			# ''p'' = 2<sup>''k''</sup> ± 1 or ''p'' = 4<sup>''k''</sup> ± 3 for some natural number ''k''.
			# 2<sup>''p''</sup> − 1 is prime (a Mersenne prime).
			# (2<sup>''p''</sup> + 1) / 3 is prime (a [[Wagstaff prime]]).

			If ''p'' is an odd [[composite number]], then 2<sup>''p''</sup> − 1 and (2<sup>''p''</sup> + 1)/3 are both composite. Therefore it is only necessary to test primes to verify the truth of the [[conjecture]].

			Currently, the known numbers for which all three conditions hold are: 3, 5, 7, 13, 17, 19, 31, 61, 127.

			The New Mersenne conjecture can be thought of as an attempt to salvage the centuries-old Mersenne's conjecture, which is false. However, according to [http://www.mersenneforum.org/showpost.php?p=53533&postcount=3 Robert D. Silverman], [[John Selfridge]] agreed that the New Mersenne conjecture is "obviously true" as it was chosen to fit the known data and counter-examples beyond those cases are exceedingly unlikely. It may be regarded more as a curious observation than as an open question in need of proving.

			Renaud Lifchitz has shown that the ''NMC'' is true for all integers less than or equal to ''16,777,212'' by systematically testing all odd primes for which it is already known that one of the conditions holds. [http://www.primenumbers.net/rl/nmc/ His website] documents the verification of results up to this number. Another, currently more up-to-date status page on the NMC is [http://primes.utm.edu/mersenne/NewMersenneConjecture.html The New Mersenne Prime conjecture.]

			== Lenstra–Pomerance–Wagstaff conjecture ==
			[[Hendrik Lenstra\|Lenstra]], [[Carl Pomerance\|Pomerance]], and [[Samuel S. Wagstaff Jr.\|Wagstaff]] have conjectured that there is an infinite number of [[Mersenne prime]]s, and, more precisely, that the number of Mersenne primes less than ''x'' is [[Asymptotic analysis\|asymptotically]] approximated by

			:<math>e^\gamma\cdot\log \log(x)/\log(2),</math><ref name=heur>http://primes.utm.edu/mersenne/heuristic.html</ref>

			where γ is the [[Euler–Mascheroni constant]].
			In other words, the number of Mersenne primes with exponent ''p'' less than ''y'' is asymptotically

			:<math>e^\gamma\cdot\log(y)/\log(2).</math><ref name=heur/>

			This means that there should on average be about <math>e^\gamma\cdot\log(10)/\log(2)</math> ≈ 5.92 primes ''p'' of a given number of decimal digits such that <math>M_p</math> is prime.

			== See also ==
			* [[Lucas–Lehmer primality test]]
			* [[Lucas primality test]]
			* [[Catalan's Mersenne conjecture]]

			== References ==

			*{{cite journal
			\| author = [[Paul T. Bateman\|Bateman, P. T.]]; [[John Selfridge\|Selfridge, J. L.]]; [[Samuel S. Wagstaff, Jr.\|Wagstaff, Jr., Samuel S.]]
			\| title = The new Mersenne conjecture
			\| journal = [[American Mathematical Monthly]]
			\| volume = 96
			\| year = 1989
			\| pages = 125–128
			\| id = {{MathSciNet \| id = 0992073}}
			\| doi = 10.2307/2323195
			\| issue = 2
			\| publisher = Mathematical Association of America
			\| jstor = 2323195 }}
			*{{cite book
			\| author = [[Leonard Eugene Dickson\|Dickson, L. E.]]
			\| title = [[History of the Theory of Numbers]]
			\| publisher = Carnegie Institute of Washington
			\| year = 1919
			\| ol=6616242M
			\| pages = 31 }} Reprinted by Chelsea Publishing, New York, 1971, ISBN 0-8284-0086-5.
			{{reflist}}

			== External links ==
			* The Prime Pages. [http://primes.utm.edu/glossary/page.php?sort=MersennesConjecture Mersenne's conjecture.]

			[[Category:Conjectures about prime numbers]]

en>RjwilmsiBot: /* Metric formulas */fixing page range dashes using AWB (7903)

2012-01-16T03:12:57Z

Metric formulas: fixing page range dashes using AWB (7903)

New page

47 year old Swimming Coach or Instructor Giovanni Koehler from Labrador City, usually spends time with hobbies and interests for instance guitar, como ganhar dinheiro na internet and papercraft. Last month just made a journey to Rwenzori Mountains National Park.<br><br>Also visit my site - [http://www.comoganhardinheiro101.com ganhando dinheiro na internet]

Disphenoid - Revision history

en>Tomruen: Not exactly, doesn't need to be regular Undid revision 641568089 by RobertLovesPi (talk)

en>Tomruen: /* Special cases and Generalizations */

en>Tomruen: /* Special cases and Generalizations */

en>RjwilmsiBot: /* Metric formulas */fixing page range dashes using AWB (7903)