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Topic: Skewes number

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	Skewes' Number
		G. Hardy once described Skewes' Number as "the largest number which has ever served any definite purpose in mathematics," though it has long since lost that distinction.
		Skewes' numbers – there are actually two of them – came about from a study of the frequency with which prime numbers occur.
		Gauss's well-known estimate of the number of prime numbers less than or equal to n, pi(n), is the integral from u=0 to u=n of 1/(log u); this integral is called Li(n).
	www.daviddarling.info /encyclopedia/S/Skewes_Number.html (346 words)

	Skewes' number - Wikipedia, the free encyclopedia
		Littlewood proved in 1914 that there is such a number (and so, a first such number); and indeed found that the sign of the difference π(x) − Li(x) changes infinitely often.
		Skewes' task was to make Littlewood's existence proof effective: exhibiting some concrete upper bound for the first sign change.
		Although both Skewes numbers are big compared to most numbers encountered in mathematical proofs, neither is anywhere near as big as Graham's number.
	en.wikipedia.org /wiki/Skewes'_number (419 words)

	Big Numbers
		Skewe's number: In 1933, Skewes used the number 10
		The number is so large that it takes a page just to describe the special notation used to write the number.
		Skewe's number and Graham's number are described in The Penguin Dictionary of Curious and Interesting Numbers by David Wells.
	thinkzone.wlonk.com /MathFun/BigNum.htm (256 words)

	Prime number theorem - Wikipedia, the free encyclopedia
		In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers.
		The prime number theorem then states that the limit of the quotient of the two functions π(x) and x/ln(x) as x approaches infinity is 1.
		Because of the connection between the Riemann zeta function and π(x), the Riemann hypothesis has considerable importance in number theory: if established, it would yield a far better estimate of the error involved in the prime number theorem than is available today.
	en.wikipedia.org /wiki/Prime_number_theorem (654 words)

	KryssTal : Introduction to Numbers
		The sum of the natural numbers is the value obtained when a selection of the numbers (beginning from 1) are added together.
		Square Numbers are integers that are the square of smaller integers.
		Skewes' Number is far, far larger than the number of all the particles in the observable Universe.
	www.krysstal.com /numbers.html (1845 words)

	Skewes' number -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-27)
		Skewes proved in 1933 that, assuming that the (Click link for more info and facts about Riemann hypothesis) Riemann hypothesis is true, there exists a number x violating π(x)
		Skewes' task was to make Littlewood's existence proof (Click link for more info and facts about effective) effective: exhibiting some concrete upper bound for the first sign change.
		Although both Skewes numbers are big compared to most numbers encountered in mathematical proofs, neither is anywhere near as big as (Click link for more info and facts about Graham's number) Graham's number.
	www.absoluteastronomy.com /encyclopedia/s/sk/skewes_number.htm (486 words)

	Read about Skewes' number at WorldVillage Encyclopedia. Research Skewes' number and learn about Skewes' number here! (Site not responding. Last check: 2007-10-27)
		In number theory, Skewes' number is by definition the smallest natural number x for which the comparison
		Littlewood proved in 1914 that there is such a number (and so, a first such number); and indeed found that the sign of the difference changes infinitely often.
		Although both Skewes numbers are big compared to most numbers encountered in mathematical proofs, neither is anywhere near as big as
	encyclopedia.worldvillage.com /s/b/Skewes'_number (385 words)

	Developing a Feel for Large Numbers Like a Googolplex
		The maximum possible number of plies (each "move" consists of two plies, one involving action by white, the other involving action by fl) in a game is bounded by 100(616 + 30 + 1) = 12700.
		Skewes' number is an upper bound on when the first change of sign must occur by, given the truth of the Riemann hypothesis.
		Skewes' number is in the range 10^^4 to 10^^5.
	www.gbbservices.com /math/large.html (2112 words)

	Large Numbers at MROB (Site not responding. Last check: 2007-10-27)
		Googolplex is a class-3 number, (although it can also be represented exactly) and all numbers within an order of magnitude of googolplex are also class-3.
		These numbers occur in the study of prime numbers, and particularly the frequency of occurrance of prime numbers.
		It is easy to see that Skewes' Number is bigger than googolplex, but not so easy to figure out which of Graham's Number and the Moser is bigger.
	home.earthlink.net /~mrob/pub/math/largenum-2.html (1551 words)

	New Scientist Back Page - Symbolic uses (Site not responding. Last check: 2007-10-27)
		Avogadro's number, the number of molecules in a mole of substance, is 6.02252 * 10
		But the largest number ever needed to solve a genuine problem (in pure mathematics) is a number called Skewes' number, which appears in a branch of maths called number theory.
		This is Graham's number, used in a part of combinatorics called Ramsey theory, and it is so large that a new number notation system had to be introduced to define it.
	www.newscientist.com /backpage.ns?id=lw77 (1309 words)

	List of numbers - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-27)
		Other numbers that are notable for their mathematical properties or cultural meanings include:
		This is a table of English names for positive rational numbers less than 1.
		Keep in mind that rational numbers like 0.12 can be represented in infinite ways, e.g.
	www.eastcleveland.us /project/wikipedia/index.php/List_of_numbers (712 words)

	Math Lair - Really Big Numbers (Site not responding. Last check: 2007-10-27)
		It is tremendously large, much larger than the number of particles in the universe or even a googolplex.
		This number was derived by R.L. Graham in 1977 from a problem in Ramsay theory (a branch of combinatorics) which concerns bichromatic hypercubes.
		Normal exponential notation is not enough to express this number, so a special "arrow" notation, which was devised by Knuth in 1976, is used to represent this number.
	www.stormloader.com /ajy/largenum.html (235 words)

	Category:Number Theory Information - Articles Free (Site not responding. Last check: 2007-10-27)
		Traditionally, number theory mathematics concerned with the properties of integers and many open problems that are easily understood even by non-mathematicians.
		More generally, the field has come to be concerned with a wider class of problems that arose naturally from the study of integers.
		Number theory may be subdivided into several fields according to the methods used and the questions investigated.
	www.articlesfree.com /index.php?title=Category:Number_theory (98 words)

	Prime Curios!: 67
		The prime number 67 is the largest known digit sum with this property (as of the twentieth century).
		The smallest prime p which divides the number of composites less than the (p + 1)th prime.
		The alphabetical value of this prime number in its Roman numeral-based representation (LXVII) is the reverse of 67.
	primes.utm.edu /curios/page.php/67.html (305 words)

	factoids > Graham's number
		Graham's number cannot be expressed using the conventional notation of powers, and powers of powers.
		Next construct the number 3^^^...^^^3 where the number of arrows is the previous 3^^^...^^^3 number.
		Now continue this process, making the number of arrows in 3^^^...^^^3 equal to the number at the previous step, until you are 63 steps, yes, sixty-three, steps from 3^^^^3.
	www-users.cs.york.ac.uk /~susan/cyc/g/graham.htm (288 words)

	Notable Properties of Specific Numbers at MROB
		This number is known to not be prime, with the smallest known factor being 316912650057057350374175801344000001 = 2
		The factorial of the single-perturbation count, a highly theoretical estimate of the number of different ways all the particles in the known universe could be randomly shuffled at each moment in time since the universe's creation.
		For a discussion of more and larger numbers, particularly those that are so large that their values are difficult to express in any form, continue to my large numbers page.
	home.earthlink.net /~mrob/pub/math/numbers-15.html (1417 words)

	[No title] (Site not responding. Last check: 2007-10-27)
		His results, which in current terms placed the number somewhere between 10^51 to 10^63, were visionary; in fact, a sphere having the radius of Pluto's orbit would contain on the order of 10^51 grains.
		For arbitrary numbers between, say, 10^150 and 10^1,000,000, ECM stands as the method of choice, although ECM cannot be expected to find all factors of such gargantuan numbers.
		The FFT reduces the number of operations down to the order of D log D. (For example, for two 1,000-digit numbers, the grammar school method may take more than 1,000,000 operations, whereas an FFT might take only 50,000 operations.) A full discussion of the FFT algorithm for multiplication is beyond the scope of this article.
	cryptome.quintessenz.at /mirror/googol.txt (3200 words)

	A table of prime counts pi(x) to 5e15
		This is an extended table of the values of pi(x), the number of primes <= x.
		Positive values of these differences indicate a deficit of primes (compared to the number predicted by the corresponding theoretical estimate); negative values indicate a surplus of primes.
		In consequence, the problem of determining N, or at least confining it within ever narrower bounds, is sometimes referred to as "Skewes' problem." R. Lehman (1966) improved the upper bound to N < 1.65e1165, and H. te Riele (1986) improved it to N < 6.69e370.
	www.trnicely.net /pi/tabpi.html (741 words)

	Philosophical Fortnights: Mathematics: items of interest
		Research on the Prime Number Theorem has yielded one of the largest numbers with a proper name: Skewes’ number: 10^(10^(10^34))).
		Skewes showed that the first n for which Li(n) > π(n) is less than the number later named after him.
		In 1955 Skewes showed that if the Riemann Hypothesis is false, then Li(n) ≥ π(n) for at least one n less than 10^(10^(10^(10^3))), which is much larger than Skewes’ number; this is called the second Skewes number.
	tlonuqbar.typepad.com /phfn/2005/06/mathematics_ite.html (886 words)

	Skewes' number - Encyclopedia Glossary Meaning Explanation Skewes' number (Site not responding. Last check: 2007-10-27)
		Skewes' number - Encyclopedia Glossary Meaning Explanation Skewes' number.
		Here you will find more informations about Skewes' number.
		The orginal Skewes' number article can be editet
	www.encyclopedia-glossary.com /en/Skewes-number.html (452 words)

	Ultrafinitism - Wikpedia (Site not responding. Last check: 2007-10-27)
		Thus some ultrafinitists will deny the existence of, for example, the floor of the first Skewes' number, which is a huge number defined using the exponential function as exp(exp(exp(79))), or
		The reason is that nobody has yet calculated what natural number is the floor of this real number, and it may not even be physically possible to do so.
		Other considerations of the possibility of avoiding unwieldily large numbers can be based on complexity theory, including the notion of feasible number.
	www.bostoncoop.net /~tpryor/wiki/index.php?title=Ultrafinitism (228 words)

	Survey 2000: Bêche-de-mer and trochus populations at Ashmore Reef: Results (Site not responding. Last check: 2007-10-27)
		Holothuria leucospilota is the most abundant species (by number) at Ashmore Reef representing in excess of 5 million individuals or 83% of the all bêche-de-mer (Figure 6).
		Large numbers of lower commercial-value bêche-de-mer were found on the reef flat (Table 3).
		Interestingly, a far higher density and species richness of low value beche-der-mer was found in the deeper habitats (shallow and deeper lagoons) compared to the shallower habitats (reef flat and crest).
	www.deh.gov.au /coasts/mpa/ashmore/survey/results.html (903 words)

	Large Numbers
		Somewhat above the googol lie numbers that present a sharp challenge to practitioners of the art of factoring: the art of breaking numbers into their prime factors, where primes are themselves divisible only by 1 and themselves.
		The idea is to use what is called modular arithmetic, which keeps the sizes of numbers under control so that machine memory is not exceeded, and adroitly scan ("sieve") over trial factors.
		To get a better sense of how enormous some numbers truly are,imagine that the 10-digit number representing the age in years of the visible universe were a single word on a page.
	www.fortunecity.com /emachines/e11/86/largeno.html (3119 words)

	Irrigation Efficiency (Site not responding. Last check: 2007-10-27)
		This embodies a number of variations from a traditional experimental study or a simple industry survey.
		Full results for the three crops are published in three reports (Skewes and Meissner, 1997a; Skewes and Meissner, 1997b; Skewes and Meissner, 1998) Data was collected from a series of sites across the Riverland (SA) and Sunraysia (NSW and Vic), by a combination of interview and field evaluation.
		A number of the case study irrigators cited irrigation system setup, age, and maintenance as limiting factors in their ability to manage irrigation as well as they would like.
	www.sardi.sa.gov.au /pages/horticulture/spn/hort_spn_irrigation.htm (4003 words)

	Thomas R. Nicely's Home Page
		Code written primarily in GNU C, and distributed asynchronously across available personal computers running under extended DOS, Windows, and GNU/Linux, is employed to enumerate primes, prime gaps, prime constellations (twins, triplets, and quadruplets) and their reciprocal sums (to extrapolate estimates for the corresponding Brun's constants).
		Following are some websites of relevance to mathematics in general, and number theory in particular.
		PARI-GP, a software package for computer-aided number theory, including the ultraprecision libpari C libraries and the gp programmable interactive calculator.
	www.trnicely.net (3790 words)

	Math Forum Discussions (Site not responding. Last check: 2007-10-27)
		number of sequences of length 10^41 (the number
		This is an upper bound on the number of branches
		Let (n,k) be the k-iterate of the number* of digits that n has.
	mathforum.org /kb/thread.jspa?messageID=371175 (2609 words)

	MATHEWS: All Numbers Large and Beautifull
		Think of a very very large number, think of the fastest possible computer, think of the age of the universe, think of the number of atoms in the universe a.s.o., its all nothing compared to the infinity.
		The total number of wheat grains placed on a chess board when you place 1 grain on square 1, 2 grains on square 2, 4 grains on square 3 a.s.o.
		The 3184th Fibonacci number is an apocalpyse number having excactly 666 digits.
	www.wschnei.de /number-theory/large-numbers.html (1196 words)

	11N: Multiplicative number theory (Site not responding. Last check: 2007-10-27)
		Huxley, M. "Dirichlet polynomials", in Elementary and analytic theory of numbers (Warsaw, 1982), 307--316, Banach Center Publ., 17, PWN, Warsaw, 1985.
		Maximum number of primes in intervals (prime constellations)
		Estimates of theta(x)=sum(log(p), p < x) and the relation to the prime number theorem.
	www.math.niu.edu /~rusin/known-math/index/11NXX.html (486 words)

	Nat' Academies Press, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003)
		(By way of contrast, the number of atoms in the cosmos is thought to have about eighty digits.) This monstrosity attained fame as “Skewes’ number,” the largest number ever to emerge naturally from a mathematical proof up to that time.
		In 1955 Skewes improved his result, this time without assuming the truth of the Riemann Hypothesis, to a number of a mere 10
		(a number, that is, of a mere 1,166 digits), and established an important general theorem about the upper bound.
	www.nap.edu /books/0309085497/html/236.html (576 words)

	Extreme Prefixes
		Computer engineers and programmers typically express numbers in powers of two because of the use of two-valued bits for computer memory and logic (each bit can represent a 0 or a 1).
		Incidentally, speaking of Googling, the name Google appears to be a variation of "googol," itself an extremely large number (suggesting Google can find information from a huge number of websites, I suppose).
		Even larger numbers (link2, link3) have been defined, such as Skewes' number, Graham's number, and the Moser, which I won't even try to describe.
	www.lewrockwell.com /orig/kinsella6.html (729 words)

	Prime Curios!: 79
		79 is the smallest number p such that: (1) p is prime; (2) p reversed is prime; (3) p + 10^120 is prime; (4) (p + 10^120) reversed is prime.
		There were 79 unprovoked shark attacks reported around the world in the year 2000, the largest number since the International Shark Attack File began compiling statistics in 1958.
		Each of the numbers 1 to 79 gives a larger number when you write out its English name and add the letters using a=1, b=2, c=3,...
	primes.utm.edu /curios/page.php/79.html (432 words)

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