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Topic: Prime number theorem

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	How many primes are there?
		The prime number theorem was stated with a=0, but it has been shown that a=1 is the best choice:
		Clearly Legendre's conjecture is equivalent to the prime number theorem, the constant 1.08366 was based on his limited table for values of pi(x) (which only went to x = 400,000).
		Gauss was also studying prime tables and came up with a different estimate (perhaps first considered in 1791), communicated in a letter to Encke in 1849 and first published in 1863.
	primes.utm.edu /howmany.shtml (1771 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.
		There is an analogue of the prime number theorem that describes the "distribution" of irreducible polynomials over a finite field; the form it takes is strikingly similar to the case of the classical prime number theorem.
	en.wikipedia.org /wiki/Prime_number_theorem (1482 words)

	Prime number - Wikipedia, the free encyclopedia
		The study of prime numbers is part of number theory, the branch of mathematics which encompasses the study of natural numbers.
		For a long time, prime numbers were thought as having no possible application outside of number theory; this changed in the 1970s when the concepts of public-key cryptography were invented, in which prime numbers formed the basis of the first algorithms such as the RSA cryptosystem or the Diffie-Hellman key-exchange algorithm.
		For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic.
	en.wikipedia.org /wiki/Prime_number (4439 words)

	PlanetMath: prime number theorem
		For a long time it was an open problem to find an elementary proof of the prime number theorem (“elementary” meaning “not involving complex analysis”).
		An elementary proof of the prime number theorem.
		This is version 16 of prime number theorem, born on 2001-10-15, modified 2006-09-05.
	planetmath.org /encyclopedia/PrimeNumberTheorem.html (224 words)

	The distribution of primes; the Prime Number Theorem (Site not responding. Last check: 2007-10-14)
		While the majority of integers are composite numbers, the Prime Number Theorem says that prime numbers occur rather frequently, distributed fairly densely among the integers.
		Another immediate consequence of the PNT is that the value of the nth prime is approximately n(ln n), for sufficiently large values of n.
		But there are infinitely many prime numbers in any arithmetic progression, as long as the initial term of the progression and the value of increment do not share a common divisor except the number 1.
	www.math.psu.edu /tseng/class/Math140H/PNT.html (412 words)

	The Prime Number Theorem (Site not responding. Last check: 2007-10-14)
		The prime numbers are distributed among the integers in a very irregular way.
		The Prime Number Theorem states that the number pi(n) of primes at most n is asymptotic to n/log(n), where log(n) is the natural logarithm of n (to the base e).
		And a rough estimate of the number of 5-digit primes, that is, of pi(100,000)-pi(10,000), is 7600 (the actual number is 8363).
	olympiads.win.tue.nl /ioi/ioi94/contest/day1prb3/pnt.html (242 words)

	Prime numbers
		Prime numbers and their properties were first studied extensively by the ancient Greek mathematicians.
		Fermat's Little Theorem is the basis for many other results in Number Theory and is the basis for methods of checking whether numbers are prime which are still in use on today's electronic computers.
		The statement that the density of primes is 1/log(n) is known as the Prime Number Theorem.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Prime_numbers.html (1600 words)

	The 1896 Proof of Prime Number Theorem
		Possibly the single most important problem in Analytic Number Theory, and certainly the driving force behind many of its' results, is the question of the distribution of the prime numbers.
		The proof, at the end of the nineteenth century, of what is now known as the Prime Number Theorem is one of mathematics' greatest achievements, representing the culmination of the work of some of history's finest mathematicians.
		The distribution of the prime numbers, like so many other areas of mathematics, was first investigated by Euclid, who proved that the number of primes is infinite.
	www.freewebs.com /history_of_mathematics (991 words)

	The Prime Machine
		A prime number is a natural number greater than 1 that can be divided without remainder only by itself and by 1.
		To the left of N (unless N=2) is the largest prime number less than N and to the right (unless we are right at the limit of the current range) is the smallest prime number greater than N.
		According to the prime number theorem, in the limit as N tends to infinity green and blue become identical.
	www.math.utah.edu /~pa/math/machine.html (1623 words)

	Prime number
		In mathematics, a prime number, or prime for short, is a natural number larger than 1 that has as its only positive divisors (factors) 1 and itself.
		In number theory itself, one talks of pseudoprimes, integers which, by virtue of having passed a certain test, are considered probable primes but are in fact composite (such as Carmichael numbers).
		Prime ideals are an important tool and object of study in commutative algebra and algebraic geometry.
	www.fastload.org /pr/Prime_number.html (1738 words)

	The Prime Number Theorem (PNT)
		The PNT as a theorem asserts that the following interpolation formulas, which attempt to smoothly fit onto the trends in the distribution of the primes, are asymptotically exact.
		The theorem is of particular interest to mathematicians in that it follows from an as-yet unproven conjecture by Riemann on the distribution of zeros of the zeta function, a complicated topic which is not required for our simple analysis here.
		Since there are an infinite number of numbers left, it is easier to think in terms of remaining blocks of numbers; after removing 2 and 3, we have blocks of 2×3=6, and in any arbitrary block of 6 consecutive numbers beyond the 3, there will be exactly 2 non-zero numbers.
	cnx.org /content/m12764/latest (1710 words)

	Several Proofs of the Twin Primes Conjecture
		are not all the twin prime intervals, i.e.,
		Because all numbers n + k and n - k are not prime numbers, equations (2)-(4) are necessary for proving the infinitude but are not sufficient for finding the existence of prime numbers at these locations.
		The PNT is strongest for providing proofs of the infinite number of primes and is weakest for accurately finding the existence of primes at given locations n.
	www.coolissues.com /mathematics/Tprimes/tprimes.htm (508 words)

	11: Number theory
		Number theory is one of the oldest branches of pure mathematics, and one of the largest.
		Elementary number theory involves divisibility among integers -- the division "algorithm", the Euclidean algorithm (and thus the existence of greatest common divisors), elementary properties of primes (the unique factorization theorem, the infinitude of primes), congruences (and the structure of the sets Z/nZ as commutative rings), including Fermat's little theorem and Euler's theorem extending it.
		Questions in algebraic number theory often require tools of Galois theory; that material is mostly a part of 12: Field theory (particularly the subject of field extensions).
	www.math.niu.edu /~rusin/known-math/index/11-XX.html (2587 words)

	The Prime Glossary: prime number theorem
		The prime number theorem implies that the probability that a random number n is prime, is about 1/log n (technically, the probability a number m chosen from the set {1,2,...,n} is prime is asympototic to 1/log n).
		It also imples that the average gap between primes near n is about log n and that log (n#) is about n (n# is the primorial function).
		It is surprisingly difficult to give a reasonable heuristic argument for the prime number theorem (and not coincidentally, the proof of this theorem is rather involved).
	primes.utm.edu /glossary/page.php?sort=PrimeNumberThm (242 words)

	Amazon.com: The Prime Number Theorem (London Mathematical Society Student Texts): Books: G. J. O. Jameson (Site not responding. Last check: 2007-10-14)
		Altogether, this textbook is outstandingly suitable for both a course on prime number theory, at the upper-graduate level, and as a source for self-instruction...
		At first glance the prime numbers appear to be distributed in a very irregular way amongst the integers, but it is possible to produce a simple formula that tells (in an approximate but well-defined sense) how many primes can be found that are less than any integer.
		This textbook introduces the prime number theorem and is suitable for advanced undergraduates and beginning graduate students.
	www.amazon.com /Number-Theorem-Mathematical-Society-Student/dp/0521891108 (1184 words)

	The Prime Number Theorem (Site not responding. Last check: 2007-10-14)
		A proof of the prime number theorem involving Fourier transforms
		P.T. Bateman, “Major figures in the history of the prime number theorem”, Abstracts of the American Mathematical Society (87th annual meeting, San Francisco), 1981, p.2.
		The theorem which we shall eventually prove is the famous theorem of Hadamard and de la Vallée Poussin..."
	www.maths.ex.ac.uk /~mwatkins/zeta/pnt.htm (233 words)

	pntdescr
		The prime number theorem is unquestionably one of the great theorems of mathematics, but it is often presented as an outlying topic of number theory because the proof is deeply analytic.
		The rationale for this book is that the prime number theorem (together with other results in the same family) merits an account in its own right.
		This follows the tradition of A.E. Ingham's classic The Distribution of Prime Numbers, but the book is at a lower level than Ingham, suitable for current undergraduate degree courses, with prerequisites kept to a minimum.
	www.maths.lancs.ac.uk /~jameson/pntdescr (203 words)

	Untitled Document
		The statement (4) is often known as "the" prime number theorem and was proved independently by
		Versions of elementary proofs of the prime number theorem appear in final section of Nagell (1951) and in Hardy and Wright (1979, pp.
		Selberg, A. "An Elementary Proof of the Prime Number Theorem." Ann.
	users.skynet.be /fa956617/math/topics/PrimeNumberTheorem.html (784 words)

	Euclid Theorem: (Site not responding. Last check: 2007-10-14)
		Since these are all possible prime numbers, M is not divisible by any prime number, and therefore M is not divisible by any number.
		That means that M is also a prime number.
		But clearly M > N, which is impossible, because N was supposed to be the largest possible prime number.
	web01.shu.edu /projects/reals/logic/proofs/euclidth.html (172 words)

	Chance News 11.02
		In practice, 200 digit prime numbers used for encryption are chosen by a method that produces numbers called "probably-primes" which are known to be prime with a very high probability.
		This apparent randomness in the distribution of prime numbers has suggested to some that, even though the primes are a deterministic sequence of numbers, we might learn about their distribution by assuming that they have, in fact, been produced randomly.
		Then, using the prime number theorem, the probability that n, n+ 2 for odd n is a twin prime is approximately 1/log(n)xlog(n+2).
	www.dartmouth.edu /~chance/chance_news/recent_news/chance_primes_chapter2.html (5555 words)

	Notes and Literature on Prime Numbers (Site not responding. Last check: 2007-10-14)
		A prime number is a natural number greater than 1 that can be divided evenly only by 1 and itself.
		A standard textbook of Number Theory, intended for use in a first course in Number Theory, at the upper undergraduate or beginning graduate level is: I. Niven, H.S. Zuckerman, H.L. Montgomery, An Introduction to the Theory of Numbers, 5th edition, Wiley, 1991.
		A major application of number theory and prime numbers is in cryptography.
	www.math.utah.edu /~alfeld/math/prime.html (321 words)

	[No title]
		A very large number of proofs can be found in the litteratures (this is certainly one of the mathematical result with so many proofs).
		Sieves can be used to study statistical data on prime numbers, like counting the number of primes, looking for small and large gaps between consecutive primes, counting the number of twin primes (primes p for which p+2 is also a prime), counting primes with a given form, etc.
		At the end of the 1930's, Cramer proposed a model [4] to construct conjectures on distribution of prime numbers.
	numbers.computation.free.fr /Constants/Primes/countingPrimes.html (1433 words)

	The Prime Number Theorem (Site not responding. Last check: 2007-10-14)
		A proof of the prime number theorem involving Fourier transforms
		P.T. Bateman, “Major figures in the history of the prime number theorem”, Abstracts of the American Mathematical Society (87th annual meeting, San Francisco), 1981, p.2.
		The theorem which we shall eventually prove is the famous theorem of Hadamard and de la Vallée Poussin..."
	secamlocal.ex.ac.uk /~mwatkins/zeta/pnt.htm (233 words)

	Abstract: A Direct Proof of the Prime Number Theorem, Dr. Stephen Lucas (Site not responding. Last check: 2007-10-14)
		Abstract: A well known theorem of number theory is the prime number theorem, which states that the number of prime numbers less than or equal to some integer x is asymptotically equal to x/log(x).
		We use this, along with the properties of a function related to the Riemann zeta- function, to form a new proof of the prime number theorem.
		The same technique is also used to find both Riemann's approximation to the number of primes and an exact summation.
	www.math.jmu.edu /~brownet/colloquia/18092006.html (369 words)

	Prime Numbers (Site not responding. Last check: 2007-10-14)
		No the number "1" is not a prime number -- by definition.
		Strictly speaking the number 1 could be considered a prime number and once was.
		Therefore with respect to "1", I include it whenever I speak of prime numbers.
	www.grandunification.com /hypertext/Prime_Numbers.html (278 words)

	Course 428 - Prime Numbers (Site not responding. Last check: 2007-10-14)
		He did not succeed in proving the Prime Number Theorem himself; he could only show that it followed from the so-called Riemann Hypothesis.
		The Prime Number Theorem (which had been postulated by Gauss) was finally proved just 100 years ago by Hadamard and de la Vallée Poussin (independently) using Riemann's method, after showing that the zeros of
		Euclid's theorem (that there are an infinity of primes).
	www.maths.tcd.ie /pub/official/Courses/428in9798.html (259 words)

	1.html
		After working through these materials, the student will understand the content of the Prime Number Theorem and will be able to use the
		is not prime then the program moves to the next line.
		There are 168 prime numbers less than 1000 and the program calculates 168/1000 =.168 and 1/ln(1000) =.1447648273.
	archives.math.utk.edu /mathtech/prime-1/15.html (311 words)

	Math 259: Introduction to Analytic Number Theory (Spring 200[2-]3)
		pnt.pdf: Conclusion of the proof of the Prime Number Theorem with error bound; the Riemann Hypothesis, and some of its consequences and equivalent statements.
		Here's a new expository paper by B. Conrey on the Riemann Hypothesis, which includes a number of further suggestive pictures involving the Riemann zeta function, its zeros, and the distribution of primes.
		Here are some tables of number fields, compiled by Henri Cohen.
	www.math.harvard.edu /~elkies/M259.02/index.html (1022 words)

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