41. Ivars Peterson's MathTrek -Goldbach's Prime Pairs There are strong grounds for believing that Goldbach s conjecture is true, Uncle Petros and Goldbach s conjecture. New York Bloomsbury Publishing. http://www.maa.org/mathland/mathtrek_8_21_00.html

Search MAA Online MAA Home Ivars Peterson's MathTrek August 21, 2000 Goldbach's Prime Pairs Like the elements in chemistry, prime numbers serve as building blocks in the mathematics of whole numbers. Evenly divisible only by themselves and one, primes are a rich source of speculative ideas that mathematicians often find simple to state but difficult to prove. The Goldbach conjecture is a prime example of such a conundrum. In a letter written in 1742 to Leonhard Euler (1707-1783), the historian and mathematician Christian Goldbach (1690-1764) expressed the belief that every integer greater than 5 is the sum of three primes. (See a copy of the letter at http://www.informatik.uni-giessen.de/staff/richstein/ca/goldbach.jpg. Euler replied, pointing out that Goldbach's statement is equivalent to the conjecture that every even integer greater than or equal to 4 is the sum of two primes. He went on to note, "that every even number is a sum of two primes, I consider an entirely certain theorem in spite of that I am not able to demonstrate it." Progress in proving the Goldbach conjecture has been slow. In the best effort to date, Chen Jing-Run proved in 1966 that beyond some large number, every even integer may be written as the sum of a prime number and a number that is either a prime or a product of two primes.

42. Goldbach's Conjecture A popular magazine announced a contest to solve Goldbach s conjecture. Goldbach s conjecture. Any even integer greater than 4 is the sum of two odd http://www.math.fau.edu/locke/Goldbach.htm

Goldbach's Conjecture A popular magazine announced a contest to solve Goldbach's Conjecture . Don't expect much enthusiasm from the mathematical community. Goldbach's Conjecture . Any even integer greater than 4 is the sum of two odd primes. Vinogradov (1937): There is an integer N such that any odd integer greater than N is the sum of three primes. Chen and Wang (1989): e e Liu and Wang (2002): e Why aren't mathematician's thrilled? Over the years, many of us have received purported proofs of famous conjectures or recently proven theorems. Examples The four colour theorem: The shortest accepted proofs so far (Haken and Appel, Seymour) have 500 or more cases. No mathematican expects that somebody will find a two-page solution in the near future. Fermat's last theorem: Andrew Wiles solved this (with a little help on one piece) after a seven-year effort. The proof is several hundred pages long. Again, no short proof is expected. Angle trisection, duplication of the cube, squaring the circle: These cannot be done with ruler and compass. It is extremely hard to convince a non-mathematician of this. However, the proof is understandable to students in undergraduate mathematics programs. If I left out your favorite problem, you don't need to contact me.

43. (Golf) GoldBach's Conjecture (Alpha-Geek.com) (Golf) GoldBach s conjecture. 200308-19 843PM There was a Perl golf contest Goldbach s conjecture, which basically statesFor all even integers http://alpha-geek.com/2003/08/19/golf_goldbachs_conjecture

(Golf) GoldBach's Conjecture There was a Perl golf contest floating about concerning Goldbach's Conjecture, which basically states:For all even integers... Navigation Main One-liners I find stupid shit funny Grip the edge of your computer desk when the second comment reads "I said yuo are teh sUxx0r!" by Anonymous Chameleon Do you like Bloglines? I do. But the number of feeds I was trying to deal with quickly got away from me. I don't think folders are the answer. Therefore, I've created this work-in-progress, which does a few cool things... Beercasting Recipe... for... disaster Chuckie , we should never engage in something like this (via Dowbrigade News OrganicHTML What the Hell is this thing? New cpansearch.perl.org (via Ask Bjorn Hansen Who has Google acquired: Google Acquisitions Sitepoint finds a new way to annoy It's true. For a site that employs people that get it , they use very invasive (and utterly useless) annoyance tactics. More One-liners RSS del.icio.us/jms18 Small ISVs:� You need Developers, not Programmers ... Working more productively with bash 2.x/3.x About About Jeremy Smith Jeremy Smith's Resume Calendar February 2005 Sun Mon Tue Wed Thu Fri Sat Archives February 2005 January 2005 December 2004 November 2004 ... February 2003 Categories cars and driving RSS 2.0

44. (2) Brief History Of Goldbach's Conjecture Goldbach s conjecture. In 1725 Goldbach became professor of mathematics and Goldbach s conjecture states. Every even positive integer greater than 3 is http://web.singnet.com.sg/~huens/gbpage02.htm

2). Brief History of Goldbach's Conjecture Goldbach's Conjecture In 1725 Goldbach became professor of mathematics and historian at St. Peterburg. Then, in 1728, he went to Moscow as tutor to Tsar Peter II. Goldbach's Conjecture states: Every even positive integer greater than 3 is the sum of two (not necessarily distinct) primes. Go back to Homepage.

45. Goldbach's Sequence And Goldbach's Conjecture the backdrop for proving Goldbach s conjecture by an as yet undiscovered method.One cannot rule out that perhaps Goldbach s conjecture is unprovable. http://web.singnet.com.sg/~huens/paper43.htm

Goldbach's Sequence And Goldbach's Conjecture by Huen Y.K. CAHRC, P.O.Box 1003, Singapore 911101 http://web.singnet.com.sg/~activweb/ Related URL-sites: http://web.singnet.com.sg/~huens/ email: huens@mbox3.singnet.com.sg (A short communication - 1st released: 18/12/97) Abstract 1. Introduction A very efficient way of weeding out unnecessary tests for noncontiguities in Goldbach's sequences, i.e. Goldbach(z), is to test only the high ends of Prime(z). This comes from a theorem on the contiguity of Odd(z)^2 in which it was proved that if the second largest odd integer is removed from Odd(z) before squaring, the resultant even integer sequence is never contiguous [11]. Since Prime(z) is a subset of Odd(z), we know that if Odd(z)^2 is not conitiguous then Prime(z)^2 of the same integer range will not be contiguous. This method is used here to extend the range of search for noncontiguous Goldbach(z) above 10^9. The method is determinstic on noncontiguities only. To determine contiguities, we still need to perform the full contiguity tests. 2. The Original Global Contiguity Tests

46. Goldbach Conjecture Research Euler replied that Goldbach s conjecture was equivalent to the statement that Euler was convinced that Goldbach s conjecture was true but was unable to http://www.petrospec-technologies.com/Herkommer/goldbach.htm

Goldbach Conjecture Research by Mark Herkommer May 24, 2004 The Conjecture... This conjecture dates from 1742 and was discovered in correspondence between Goldbach and Euler. It falls under the general heading of partitioning problems in additive number theory. Goldbach made the conjecture that every odd number > 6 is equal to the sum of three primes. Euler replied that Goldbach's conjecture was equivalent to the statement that every even number > 4 is equal to the sum of two primes. Because proving the second implies the first, but not the converse, most attention has been focused on the second representation. The smallest numbers can be verified easily by hand: Of course all the examples in the world do not a proof make. Research On The Conjecture... As a partitioning problem it is worth noting that as the numbers get larger the number of representations grows as well: This would suggest that the likelihood of finding that exceptional even number that is not the sum of two primes diminishes as one searches in ever larger even numbers. Euler was convinced that Goldbach's conjecture was true but was unable to find any proof (Ore, 1948). The first conjecture has been proved for sufficiently large odd numbers by Hardy and Littlewood (1923) using an "asymptotic" proof. They proved that there exists an n0 such that every odd number n > n0 is the sum of three primes. In 1937 the Russian mathematician Vingradov (1937, 1954) again proved the first conjecture for a sufficiently large, (but indeterminate) odd numbers using analytic methods. Calculations of n0 suggest a value of 3^3^15, a number having 6,846,169 digits (Ribenboim, 1988, 1995a).

47. Conjecture 1. Goldbach's Conjecture Goldbach s conjecture. In a letter of 1742 to Euler, Goldbach expressed thebelief that ‘Every integer N 5 is the sum of three primes’. http://www.primepuzzles.net/conjectures/conj_001.htm

Conjectures Conjecture 1. Goldbach's Conjecture "In a letter of 1742 to Euler, Goldbach expressed the belief that ‘ . Euler replied that this is easily seen to be equivalent to the following statement (Ref. 1, p. 291) Then as we can see the original idea was from Goldbach but the simplification and limitation of it came from Euler. By the above reasons the original statement of the Goldbach’s conjecture now is known as "the odd Goldbach conjecture". Samuli Larvala send today (11/08/98) the following interesting information about the status of the work done over this conejcture: " Matti Sinisalo has checked the conjecture up to 4*10^11. His paper was published in Math.Comp. "M.K. Sinisalo, Checking the Goldbach conjecture up to 4*10^11, Math. Comp. 61 (1993)". J-M. Deshouillers and Herman te Riele have recently checked it up to 10^14. They published a preview paper on their work when they had reached 10^13. This paper can be found at: ftp://ftp.cwi.nl/pub/herman/Goldbach/gold13.ps

48. Goldbach's Conjecture - Enpsychlopedia Goldbach s conjecture. (Redirected from Goldbach conjecture). In mathematics,Goldbach s conjecture is one of the oldest unsolved problems in number theory http://psychcentral.com/psypsych/Goldbach_conjecture

home resource directory disorders quizzes ... support forums Advertisement ( Goldbach's conjecture (Redirected from Goldbach conjecture In mathematics, Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics . It states: Every even number greater than 2 can be written as the sum of two primes . (The same prime may be used twice.) For example, etc. Contents showTocToggle("show","hide") 1 Origins 2 Heuristic justification 3 Rigorous results 4 Trivia ... edit Origins In , the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler in which he proposed the following conjecture: Every integer greater than 2 can be written as the sum of three primes. He considered 1 to be a prime number , a convention subsequently abandoned. So today, Goldbach's original conjecture would be written: Every integer greater than 5 can be written as the sum of three primes. Euler, becoming interested in the problem, answered with an equivalent version of the conjecture: Every even number greater than 2 can be written as the sum of two primes. The former conjecture is today known as the "ternary" Goldbach conjecture, the latter as the "strong" Goldbach conjecture. The conjecture that all odd numbers greater than 9 are the sum of three odd primes is called the "weak" Goldbach conjecture. Both questions have remained unsolved ever since, although the weak form of the conjecture is much closer to resolution than the strong one.

49. Flak Magazine: Review Of Uncle Petros & Goldbach's Conjecture, 05-25-00 With Uncle Petros Goldbach s conjecture, firsttime author Apostolos While Goldbach s conjecture has been shown to be true for very large even http://flakmag.com/books/goldbach.html

BOOKS Index Page Archives Submissions RECENTLY IN BOOKS Killing Yourself to Live by Chuck Klosterman The End of Poverty by Jeffrey Sachs Ice Haven by Daniel Clowes You Remind Me of Me by Dan Chaon A Long Way Down by Nick Hornby We and GWB edited by Nikki Lee and David Nett Bookmark Now edited by Kevin Smokler Haunted by Chuck Palahniuk The Turkish Gambit by Boris Akunin Witness: For the Prosecution of Scott Peterson by Amber Frey ABOUT FLAK About Flak Archives Letters to Flak Submissions ... Rejected! ALSO BY FLAK Flak Sunday Comics The Spam Blog The Remote Flak Print [6mb PDF] SEARCH FLAK flakmag.com www Powered by Google MAILING LIST Sign up for Flak's weekly e-mail updates: Subscribe Unsubscribe by Apostolos Doxiadis Bloomsbury USA "One of life's failures." That is the casual dismissal Doxiadis' narrator receives when he asks his father about his reclusive uncle Petros Papachristos, who seems to spend all of his spare time gardening at his home in Ekali, a hamlet on the outskirts of Athens. (The novel is translated from Greek and was published in Greece in 1992.) Naturally, the narrator becomes curious, deciding to find out for himself the information his father fails to provide. Fortunately for us, he takes his reader along, as he pokes around the reclusive Petros's house and craftily reroutes and intercepts his uncle's mail.

50. Goldbach's Conjecture Goldbach s conjecture. Next Twin primes conjecture Up Unsolved ProblemsPrevious Collatz Problem. Goldbach s conjecture. This conjecture claims that http://db.uwaterloo.ca/~alopez-o/math-faq/node62.html

Next: Twin primes conjecture Up: Unsolved Problems Previous: Collatz Problem Goldbach's conjecture This conjecture claims that every even integer bigger equal to 4 is expressible as the sum of two prime numbers. It has been tested for all values up to by Sinisalo. Alex Lopez-Ortiz Mon Feb 23 16:26:48 EST 1998

51. Article About "Goldbach's Conjecture" In The English Wikipedia On 24-Apr-2004 The Goldbach s conjecture reference article from the English Wikipedia on 24Apr- In mathematics, Goldbach s conjecture is one of the oldest unsolved http://fixedreference.org/en/20040424/wikipedia/Goldbach's_conjecture

The Goldbach's conjecture reference article from the English Wikipedia on 24-Apr-2004 (provided by Fixed Reference : snapshots of Wikipedia from wikipedia.org) Goldbach's conjecture In mathematics, Goldbach's conjecture is one of the oldest unsolved problemss in number theory and in all of mathematics . It states: Every even number greater than 2 can be written as the sum of two primes . (The same prime may be used twice.) For example, etc. Table of contents showTocToggle("show","hide") 1 Origins 2 Results 3 Trivia 4 External links Origins In 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler in which he proposed the following conjecture: Every number greater than 5 can be written as the sum of three primes. Euler, becoming interested in the problem, answered with a stronger version of the conjecture: Every even number greater than 2 can be written as the sum of two primes. The former conjecture is known today as the 'weak' Goldbach conjecture , the latter as the 'strong' Goldbach conjecture. (The strong version implies the weak version, as any odd number greater than 5 can be obtained by adding 3 to any even number greater than 2). Without qualification, the strong version is meant. Both questions have remained unsolved ever since. Results Goldbach's conjecture has been researched by many number theorists. The majority of mathematicians believe the (strong) conjecture to be true, mostly based on statistical considerations focusing on the

52. Goldbach's Conjecture Goldbach s conjecture is that for all even N 2, N is the sum of two primes.This would imply that for all n 1, there exists a P2 (pq, p and q prime) as n2 http://www.users.globalnet.co.uk/~perry/maths/goldbach/goldbach.htm

Goldbach's conjecture [Home] [Other Maths] Goldbach's conjecture - k , as let N=2n, then p=n-k and q=n+k gives this. This proof uses n to generate a covering system of congruences which eliminate all composites from the set of equations N - p i = x i , where p i covers the odd primes from 3 to n and we aim to prove that at least one x i is composite. From the congruences the pigeon hole principle is used to prove Goldbach's conjecture. Proof The idea of a congruence is due to Gauss. Modern day notation writes x mod y, and this indicates a number of the form x + ky, where k is an integer. For example 3 mod 7 includes -11, -4, 3, 10, 17 and so on. The integers are consider congruent mod 7. If we consider only the odd primes that are less than or equal to the square root of n, then we may express N in terms on it's congruence's to these primes. This set of primes is labelled CP for congruence primes. For example if N=28, then we consider the primes 3 and 5. 28 is 1 mod 3 and 3 mod 5. Now it is clear that if an element of TP is of the same congruence to N mod an element of CP, then N - p i cannot be prime.

53. UNCLE PETROS & GOLDBACH'S CONJECTURE: A NOVEL OF MATHEMATICAL OBSESSION This socalled Goldbach s conjecture (slightly restated from its Goldbach sconjecture demanded him whole his body, his soul and all of his time. http://www.acsu.buffalo.edu/~insrisg/bookmarks/bk01/0329petros.htm

A Novel of Mathematical Obsession (This column was first published in the March 29, 2001 ArtVoice of Buffalo.) Mathematicians are very different from the rest of us and even from other members of the scientific research community. I state that from personal experience, having dealt with many of them over a lifetime of work in an activity parallel to but never intersecting theirs. Theirs - and I speak here of world class mathematicians - is an activity so different from that of the rest of us that it is extremely difficult to gain insights into it. Greek author Apostolos Doxiadis has, in (Bloomsbury, 2000), achieved the near impossible. He gives the intelligent outsider a view of what it is like to be on the cutting edge of mathematical research. That he does so in a charming family story, a pleasant tale that holds together the mathematical insights is a further achievement of high order. Here, for example, is how the story begins: "Every family has its black sheep in ours it was Uncle Petros. "My father and Uncle Anargyros, his two younger brothers, made sure that my cousins and I should inherit their opinion of him unchallenged.

54. MathFiction: Uncle Petros And Goldbach's Conjecture (Apostolos Doxiadis) Uncle Petros and Goldbach s conjecture (1992). Apostolos Doxiadis fictionalmathematician Petros Papachristos to prove Goldbach s conjecture (that every http://math.cofc.edu/faculty/kasman/MATHFICT/mfview.php?callnumber=mf15

55. Goldbach's Conjecture Goldbach s conjecture. © Copyright 2000, Jim Loy. The modern version of Goldbach sconjecture (called Goldbach s Strong conjecture) is this http://www.jimloy.com/number/goldbach.htm

Return to my Mathematics pages Go to my home page Goldbach's Conjecture The modern version of Goldbach's Conjecture (called Goldbach's Strong Conjecture) is this: Every even number greater than 2 is the sum of two primes. Let's try a few: The conjecture is looking safe so far. Not only is each even number the sum of two primes, but the number of pairs of primes tends to increase. This trend seems to continue. But no one has ever proved that this goes on forever. All of the even number up to 400,000,000,000 have been tested, so far, with no exceptions found. Mathematicians have achieved some results in their efforts to prove (or disprove) this conjecture. In 1966, J. R. Chen showed that every sufficiently large even number is either the sum of two primes or of a prime and a near prime. A near prime is a number that is the product of two primes, like 91=7x13 or 4=2x2. No one knows just how large "sufficiently large" is. There is another Goldbach Conjecture, that every odd number greater than 5 is the sum of three primes. This is known as the Weak Goldbach Conjecture. This too has not been proved or disproved. It has been shown that if there are exceptions, then there are only a finite number of exceptions. A slightly different form of these conjectures was originally posed by Christian Goldbach, in 1742. Incidentally, if either Goldbach Conjecture is ever proven, then that would also prove that there are infinitely many primes. But we already knew that. See

56. In Quest Of Information About Goldbach's Conjecture In quest of information about Goldbach s conjecture. Aroused by an essay in NYTimes of end April 2000 I was inspired to the following arrangement http://www.private.org.il/goldbach.html

Moledet, 17.VI.2000 In quest of information about Goldbach's conjecture Aroused by an essay in N.Y. Times of end April 2000 I was inspired to the following arrangement : Put in a horizontal row the odd primes in their natural order, the same in a vertical column and put at the points of intersection of coordinates the sum of the two primes concerned. In the main diagonal we will find the double of every prime in natural order, dividing between two mirrored parts; say, in the lower, left part, we find all the possible sums, and we count their appearances according to order. The total of all splittings up to some limit is equal to the area concerned, and therefore the average of representations of every even sum will grow indefinitely as the number of primes, specially here from i to , but naturally there are fluctuations between local maxima and local minima. In the domain under examination (up to 560) these fluctuations stay between twice and half the average about x1.6 and x0.6; statistically there is no tendency for the minimum to be , contrary to Goldbach's conjecture.

57. American Mathematical Monthly, The: Wild Numbers / Uncle Petros And Goldbach's C Uncle Petros and Goldbach s conjecture, by Apostolos Doxiadis, Bloomsbury, The problem is Goldbach s conjecture and the boy eventually admits defeat 2. http://www.findarticles.com/p/articles/mi_qa3742/is_200011/ai_n8927541

@import url(/css/us/style1.css); @import url(/css/us/searchResult1.css); @import url(/css/us/articles.css); @import url(/css/us/artHome1.css); Home Advanced Search IN free articles only all articles this publication Automotive Sports FindArticles American Mathematical Monthly, The Nov 2000 Content provided in partnership with 10,000,000 articles Not found on any other search engine. Featured Titles for ASA News ASEE Prism Academe African American Review ... View all titles in this topic Hot New Articles by Topic Automotive Sports Top Articles Ever by Topic Automotive Sports Wild Numbers / Uncle Petros and Goldbach's Conjecture, The American Mathematical Monthly, The Nov 2000 by Kennedy, Steve Save a personal copy of this article and quickly find it again with Furl.net. It's free! Save it. The Wild Numbers, by Philibert Schogt, Four Walls Eight Windows, New York, 2000, 160 pp., $18.00. Uncle Petros and Goldbach's Conjecture, by Apostolos Doxiadis, Bloomsbury, New York, 2000, 209 pp., $23.95. Obsession leads to loss, despair, and tragedy. Such, at least, is the lesson of literature. Obsession has a long and distinguished literary thematic career romantic and/or sexual obsession tops the list, of course, but obsession with power or fame or money or large white cetaceans have all had their day. Obsession with science and knowledge has also, perhaps not quite so frequently, been featured in novels, Mary Shelley's Frankenstein being the archetype. The common thread in all of these novels-I can think of no counterexample-is that yielding to an obsession leads to disaster.

58. American Mathematical Monthly, The: Crystallographic Restriction, Permutations, crystallographic restriction, permutations, and Goldbach s conjecture, The Recall that Goldbach s conjecture asserts that every even natural number x http://www.findarticles.com/p/articles/mi_qa3742/is_200303/ai_n9229727

@import url(/css/us/style1.css); @import url(/css/us/searchResult1.css); @import url(/css/us/articles.css); @import url(/css/us/artHome1.css); Home Advanced Search IN free articles only all articles this publication Automotive Sports FindArticles American Mathematical Monthly, The Mar 2003 Content provided in partnership with 10,000,000 articles Not found on any other search engine. Featured Titles for ASA News ASEE Prism Academe African American Review ... View all titles in this topic Hot New Articles by Topic Automotive Sports Top Articles Ever by Topic Automotive Sports crystallographic restriction, permutations, and Goldbach's conjecture, The American Mathematical Monthly, The Mar 2003 by Bamberg, John Cairns, Grant Kilminster, Devin Save a personal copy of this article and quickly find it again with Furl.net. It's free! Save it. 1. INTRODUCTION. The object of this paper is to make an observation connecting Goldbach's conjecture, the crystallographic restriction, and the orders of the elements of the symmetric group. First recall that for an element g of a group G the order Ord(g) of g is defined to be the smallest natural number such that g^sup Ord(g)^ = id if such a number exists, and Ord(g) = [infinity] otherwise. In dimension n, the crystallographic restriction (CR) is the set Ord^sub n^ of finite orders realized by n x n integer matrices: Continue article Advertisement which should be compared to the standard formula [straight phi](m) = [product operator]^sub i^ [straight phi](P^sup r^sub i^^^sub i^). Then the CR in dimension n is given by (see [29], [30]):

59. Two Approaches To Proving Goldbach's Conjecture Two approaches to proving Goldbach s conjecture. Bernard Farley. Goldbach s conjecturestates that all even numbers greater than 2 can be expressed as the http://www.math.vt.edu/people/linnell/Ugresearch/farley.html

Two approaches to proving Goldbach's conjecture Bernard Farley Goldbach's conjecture states that "all even numbers greater than 2 can be expressed as the sum of two primes". An equivalent statement of Goldbach's conjecture was found to be "for every integer n greater than or equal to 2, there exists an integer j such that n+j and n-j are prime numbers". The purpose of the research was to approach Goldbach's conjecture using this equivalent statement. It resulted in two conjectures concerning Goldbach's conjecture. One involved a specially created sequence and another involved a lower bound that was found using counting methods.

60. Goldbach's Conjecture The conjecture merely referred to as Goldbach s conjecture is the strong onewhich is Chris Caldwell Goldbach s conjecture, part of the Prime Pages http://www.teachersparadise.com/ency/en/wikipedia/g/go/goldbach_s_conjecture.htm

Free Teacher Resources First Time Visitors Gift Certificates Education Directory ... Edit this page Goldbach's conjecture Goldbach's Conjecture is one of the oldest unsolved problems in number theory and in all of mathematics . It states: Every even number greater than 2 can be written as the sum of two primes (The same prime may be used twice.) The conjecture had been known to Descartes . The following statement is equivalent and is the one originally conjectured in a letter written by Goldbach to Euler in Every number greater than 5 can be written as the sum of three primes. . The majority of mathematicians believe the conjecture to be true, mostly based on statistical considerations focusing on the probabilistic distribution of prime numbers : the bigger the even number, the more "likely" it becomes that it can be written as a sum of two primes. We know that every even number can be written as the sum of at most six primes. As a result of work by Vinogradov, every sufficiently large even number can be written as the sum of at most four primes. Vinogradov proved furthermore that almost all even numbers can be written as the sum of two primes (in the sense that the fraction of even numbers which can be so written tends towards 1). In

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