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Topic: Proof that the sum of the reciprocals of the primes diverges

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	Number - Encyclopedia, History, Geography and Biography
		The first existence proofs of irrational numbers is usually attributed to Pythagoras, more specifically to the Pythagorean Hippasus of Metapontum, who produced a (most likely geometrical) proof of the irrationality of the square root of 2.
		Euclid devoted one book of the Elements to the theory of primes; in it he proved the infinitude of the primes and the fundamental theorem of arithmetic, and presented the Euclidean algorithm for finding the greatest common divisor of two numbers.
		Other results concerning the distribution of the primes include Euler's proof that the sum of the reciprocals of the primes diverges, and the Goldbach conjecture which claims that any sufficiently large even number is the sum of two primes.
	www.arikah.net /encyclopedia/Number (3724 words)

	Harmonic series (mathematics) - Wikipedia, the free encyclopedia
		(This proof, due to Nicole Oresme, is a high point of medieval mathematics.) Even the sum of the reciprocals of the prime numbers diverges to infinity (although that is much harder to prove; see proof that the sum of the reciprocals of the primes diverges).
		The reason is that the sum is approximated by the integral
		If p > 1 then the sum of the series is ζ(p), i.e., the Riemann zeta function evaluated at p.
	en.wikipedia.org /wiki/Harmonic_series_(mathematics) (296 words)

	Proof that the sum of the reciprocals of the ... - Wikipedia, the free encyclopedia
		Proof that the sum of the reciprocals of the...
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	en.wikipedia.org /wiki/Proof_that_the_sum_of_the_reciprocals_of_the_... (212 words)

	All Information of Prime number (Site not responding. Last check: 2007-10-21)
		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.
		With this definition, the primes of the field Q of rational numbers are represented by the standard absolute value function (known as the "infinite prime") as well as by the p-adic number on Q, for every prime number p.
		357686312646216567629137 is the largest prime number that is "left-truncatable" in decimal representation, meaning that the number, as well as all the numbers obtained by successively removing the first digit at the left of the number are prime: 357686312646216567629137, 57686312646216567629137, 7686312646216567629137,..., 9137, 137, 37 and 7 are all prime.
	prime.number.en.xvip.org (4390 words)

	There are infinitely many primes, but, how big of an infinity?
		prime number theorem, which states the number of primes less than x is approximately x/log x (the natural log), gives perhaps the best answer.
		Nevertheless, it has been proven the sum of the reciprocals of the twin primes is about is 1.90216054...(this is called Brun's constant).
		It is interesting to note that there is a good bound for the partial sums of the reciprocals of the primes.
	primes.utm.edu /infinity.shtml (1018 words)

	Reciprocals (Site not responding. Last check: 2007-10-21)
		A "large" subset of positive integers is one whose sum-of-reciprocals diverges, whereas a "small" subset of positive integers has a convergent sum-of-reciprocals.
		Answer: the primes are large -- the sum of reciprocals of primes diverges.
		The sum of reciprocals of numbers that can be written without using digit d, base r, converges.
	mcraefamily.com /MathHelp/BasicNumberPrimeReciprocals.htm (494 words)

	nrich.maths.org::Mathematics Enrichment::NRICH
		Proof: assume a to be in the interval 0 to 1.
		In it, I'm summing from N+1 to 2N, for all N, and assuming this is the same as summing all N. Please ignore it.
		The proof of this is similar to the proof that the sum of reciprocals of primes diverges.
	www.nrich.maths.org.uk /askedNRICH/edited/4016.html (2038 words)

	Leonhard Euler Encyclopedia Articles @ 216.92.11.26 () (Site not responding. Last check: 2007-10-21)
		It was reported by Legendre that often he would write down a complete mathematical proof between the first and the second call for supper though this story must be second hand, if not apocryphal.
		He investigated primitive roots, found new large primes, and deduced the infinitude of the primes from the divergence of the harmonic series.
		Euler Proved that the sum of the reciprocals of the primes diverges.
	216.92.11.26 /encyclopedia/Leonhard_Euler (1807 words)

	Meissel-Mertens constant - Free net encyclopedia
		Here γ is the famous Euler-Mascheroni constant, which has a similar definition involving a sum over all integers (not just the primes).
		The fact that there are two logarithms (log of a log) in the limit for the Meissel-Mertens constant may be thought of as a consequence of the combination of the prime number theorem and the limit of the Euler-Mascheroni constant.
		In mathematics literature it is also sometimes referred to as Kronecker's constant, or the Hadamard-de la Vallée-Poussin constant, or the prime reciprocal constant.
	www.netipedia.com /index.php/Meissel-Mertens_constant (185 words)

	Read This: Topics in the Theory of Numbers
		The properties are then used to establish results involving Diophantine Approximation and sums of squares (similar to Burger’s chapters on the Geometry of Numbers in Exploring the Number Jungle).
		The notion of Sidon sequences (sequences of nonnegative integers with the property that all pairwise sums of elements are distinct).
		The standard results concerning sums of squares (when is n a sum of two squares, a sum of three squares, or a sum of four squares) are proven.
	www.maa.org /reviews/erdossuranyi.html (1957 words)

	MCS Colloquium
		For over 250 years, mathematicians have been using analytic functions to prove theorems about the distribution of primes.
		We will discuss Euler's proof that the sum of the reciprocals of the primes diverges, the prime number theorem and its connection to the Riemann Hypothesis, and a specific instance of Dirichlet's theorem on primes in progressions.
		We will also present a calculation "a la Stark" which forms a tie between derivatives of L-functions and algebraic number theory.
	www.sci.csuhayward.edu /mathcs/colloq/2005.02.25.html (85 words)

	:::► Letter P Dictionary of Meaning www.dictionary-of-meaning.com ◄::: (Site not responding. Last check: 2007-10-21)
		Proofs of Fermat's theorem on sums of two squares
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		Proof that Nazis deliberately killed six million Jews
	www.dictionary-of-meaning.com /P_1245.html (56 words)

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