| The_Stepladder_Proof_of_The_Goldbach_Conjecture A proposed proof offered for criticism. |
| XGC An eXtension of the Goldbach Conjecture. Mathematica code. |
| Advanced_Placement_Digital_Library_for_Biology,_Physics_and_Chemistry An NSF funded Rice University Digital Library project that hosts free reviewed online resources, linked to the content outline, for AP and Pre-AP teachers and students of biology, physics and chemistr |
| All_Science_Den_com Articles about assorted topics, such as molecular biology, physics. Some include Flash animations to help explain concepts. |
| American_Communications_Foundation_Newsource__Science_&_Technology Assists the commercial broadcast news media in its coverage of science and technology, which is often under-reported because of staffing and budgetary restraints. Enhances the coverage by providing re |
| Animonautes Production of 3D media to offer learning experiences in medicine, biology, astronomy and technology. Page includes sample videos. |
| ANOVA_Science_Education Services public and private school districts and schools on the continental USA and in Hawaii through teacher and administrator seminars and workshops. |
| Asia_Link_program__MUMA_project A training project dedicated to improve the quality of higher education for heritage management in Asia and Europe. The project runs from March 2004 to August 2006. |
| Ask_Dr__Jekyl_and_Mr__Hyde Answers to a variety of science and technology questions, including questions you never thought to ask. |
| Athena NASA page features earth and space science resources for K-12 and teachers. Included are class exercises with data collecting and background information for each topic, as well as related links. |
| Attaining_Excellence_Through_TIMSS Resources for learning about and discussing the Third International Mathematics and Science Study, including related reports, data and commentaries. |
| BBC_Science_&_Nature Information about humans, animals, space, the planet Earth and various hot topics. Includes TV listings, Listen Again online radio, news reports, quizzes, picture galleries and games. |
| Berkshire_Biological_Supply_Company Online sale of living organisms for biology class, science fair projects, home teaching. |
| BETR_World_Science_and_Computer_Education Offering information on classes and camps for science and computer education in Maryland and Virginia. |
| Beyond_Discovery A series of articles that trace the origins of important recent technological and medical advances. Each story reveals the crucial role played by basic science. |
| Bill_Nye_the_Science_Guy Online science laboratory. Includes learning activities and show information. |
| Brainium A curriculum-based, online environment where teachers and students can share ideas, experiments and activities. |
| The_Bubblesphere All about soap bubbles. Bubble blowing, solutions, history, fun, Java games, bubble machines and trivia. |
| Candis_Mitchell\'s_Science_Teacher\'s_Site Designed to help educators search for internet material appropriate for their classrooms. |
| Cornell_Theory_Center_Math/Science_Gateway Links to resources in mathematics and science for educators and students in grades 9-12. |
| Cute_Science Learning materials for include coloring books, audio lectures, videos and workbooks. Subjects covered are math, science, and HTML. Includes product list and online order form. |
| Department_of_Energy_Science_Education_Programs Internship and fellowship opportunities for undergraduate students and teachers related to science, technology, engineering, and mathematics. |
| Did_You_Ever_Wonder? Each month a dozen questions are posed and answered by lab scientists on various topics on how things work in the natural world. |
| Dive_and_Discover__Expeditions_to_the_Seafloor An interactive distance learning web site designed to immerse your students in the exciting process of deep-sea research and exploration. |
| Dr__Carlson\'s_Science_Theater Video podcast of cool science demonstrations. |
| Dr__Fred\'s_Place Children's science author, Dr. Fred Bortz, including biographical information, list of books, FAQ's, and information about school visits. |
| Ed_Quest Includes guided links for students, lesson plans and references for teachers, as well as collaborative projects directed towards the middle school level. |
| Education_Links_-_Top_rated_educational_sites_ Collection of educational resources separated into categories such as astronomy, lesson plans, math, and webquests. |
| Educational_Resources_Catalog In addition to the catalog itself, links are provided to curriculum materials, professional development, software, visual and audiovisual classroom support, print material and online resources. |
| Elementary_Science_Program Includes activity of the month for students and teachers of elementary science, as well as an archive of past activities, and resources. |
| The_ENC_Digital_Dozen Each month the Eisenhower National Clearinghouse selects a dozen useful math and science sites for teachers and students. Includes archives. |
| EOA_Scientific_Internet_Campus Find educational information and interactive multimedia resources in earth science,geology,oceanography,space science, astronomy, physical science, and remote sensing. |
| ERIC_Clearinghouse_for_Science,_Mathematics,_and_Environmental_Education Goal is to provide access to information for teaching and learning about science, mathematics, and the environment. |
| Every_Child_a_Scientist For those who want to take an active role in improving the science program in their schools. |
| Extreme_Science Science news and activities with an emphasis on extremes. |
| Field_Trips Includes virtual science tours on nature topics. Includes related educational resources. |
| FizzBang_Science Hands-on science books written by a high school teacher. Includes online shopping and sample experiments. |
| Frank_Potter\'s_Science_Gems Resources are sorted by category and grade level. |
| Free_Science_Worksheets Features over science labs for biology, chemistry, and physics, and numerous other printable note skeletons, graphic organizers, and worksheets. |
| Fun_With_Science A professional geophysicist maintains this collection of resources for classroom demonstrations and hands-on exercises about physics, earth science, earthquakes, math and fun illusions. Includes video |
Glossary: IndexSearchHome Previous Next RandomMail Editor Prime Pages: Home Search Index FAQ Top 5000: Primes Provers Curios 
Goldbach wrote a letter to Euler dated June 7, 1742 suggesting (roughly) that every even integer is the sum of two integers p and q where each of p and q are either one or odd primes. Now we often word this as follows:
Goldbach's conjecture: Every even integer
n greater than two is the sum of two primes.
This is easily seen to be equivalent to
Every integer
n greater than five is the sum of three primes.
There is little doubt that this result is true, as
Euler replied to Goldbach:
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 has been made on this problem, but slowly--it
may be quite awhile before the work is complete.
For example, it has been proven that every even integer is the sum of at most six primes (Goldbach suggests two) and in 1966 Chen proved every sufficiently large even integer is
the sum of a prime plus a number with no more than two prime factors (a P2).
Vinogradov in 1937 showed that every sufficiently large odd integer can be written as the sum of at most three primes, and so every sufficiently large integer is the sum of at most four primes. One result
of Vinogradov's work is that we know Goldbach's theorem
holds for almost all even integers.
Various heuristic estimates are avaliable for the number of solutions there should be to 2n = p + q (with p, q prime). And these of course grew in size with n.
Recently, Jean-Marc Deshouillers, Yannick Saouter, and Herman te Riele have verified this up to 1014 with the help of a Cray C90 and various workstations.
If we will accept the Riemann Hypothesis, then this is enough to prove the odd Goldbach conjecture: Every odd integer greater than five is the sum of three primes.
Descartes also was aware of the two prime version of Goldbach's conjecture before Goldbach was. So is it misnamed? Erdös said that it "is better that the conjecture be named after Goldbach because, mathematically speaking, Descartes was infinitely rich and Goldbach was very poor."
When verifying the Goldbach conjecture for n
we quickly see that it is very easy to find many primes
which add to n. In 1923 Hardy and Littlewood
took the first major step toward the proof of the Goldbach
conjectures using their circle method [HL23]. Among other things, they conjectured that the number of ways of writing n as the sum of two primes, G(n), is asymptotic to twice the twin prime constant times
n/(log n)2 times the product of (p-1)/(p-2) taken over the prime divisors p of n.
That is a lot of solutions for a large n so there will be a solution with one of the prime quite small. For example, when checking all n up to 100,000,000,000,000, Richstein [Richstein2000] found the smaller prime is never larger than 5569 used in the following partition:
389965026819938 = 5569 + 389965026814369.
It was conjectured [GRL89] that the smallest prime in a
partition of n might be bounded by a constant times
(log n)2 log log n. Richstein's work shows that the constant 1.603 (from n=6) sufficies for n < 4.1014.See Also: OddGoldbachConjectureRelated pages (outside of this work) Prime conjecturesReferences:DRS98 J. M. Deshouillers, H. J. J. te Riele and Y. Saouter, New experimental results concerning the Goldbach conjecture. In "Proc. 3rd Int. Symp. on Algorithmic Number Theory," Lecture Notes in Computer Science volume 1423, pp. 204--215, 1998. MR 2000j:11143FR89 Fliegel, H. F. and Robertson, D. S., "Goldbach's comet: the numbers related to Goldbach's conjecture," J. Recreational Math., 21:1 (1989) 1--7.GRL89 A. Granville, H. J. J. te Riele and J. van de Lune,"Checking the Goldbach conjecture on a vector computer" in Number theory and its applications. R. A. Mollin editor, Kluwer, Dordrect, pp. 423--433, 1989.HL23 G. H. Hardy and J. E. Littlewood, "Some problems of `partitio numerorum' : III: on the expression of a number as a sum of primes," Acta Math., 44 (1923) 1-70. Reprinted in "Collected Papers of G. H. Hardy," Vol. I, pp. 561-630, Clarendon Press, Oxford, 1966.Ramare95 O. Ramaré, "On Schnirelmann's constant," Ann. Sc. Norm. Super. Pisa, 22:4 (1995) 645-706. MR 97a:11167Ribenboim95 P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, New York, NY, pp. xxiv+541, ISBN 0-387-94457-5. 1995. MR 96k:11112 [An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]Richstein2000 J. Richstein, "Verifying the goldbach conjecture up to 4· 1014," Math. Comp., 70:236 (2001) 1745--1749. MR 2002c:11131 (Abstract available)Sinisalo93 M. Sinisalo, "Checking the Goldbach conjecture up to 4· 1011," Math. Comp., 61:204 (1993) 931--934. MR 94a:11157Wang84 Y. Wang editor, Goldbach conjecture, Series in Pure Mathematics volume 4, World Scientific, Singapore, pp. xi+311, ISBN 9971-966-08-5; 9971-966-09-3. 1984. MR 87h:11102 (Annotation available) Chris Caldwell © 1999-2009 (all rights reserved)