Millennium Prize Problems

Goldbach's Conjecture: $1,000,000 challenge

Double Bubble Conjecture Proved

Novo recorde: GIMPS Finds Its Fourth Prime!!!!

Robert J. Harley's Group Solves Elliptic Curve Cryptosystem Exercise

Leibniz's 333-year-old problem solved

Leilão do palimpsesto com

Fields Medalists / Nevanlinna Prize 1998

ten consecutive primes in arithmetic progression

37th Known Mersenne Prime Discovered!!!

New Math. Record: primes in arithmetic progression

BEAL'S CONJECTURE

Falso alarme: CARMICHAEL'S CONJECTURE

FILIP SAIDAK PROVES CARMICHAEL'S CONJECTURE

New Amicable Pair record

2^2976221-1 is the 36th known Mersenne prime

Erdos Numbers update

TIMSS - Executive Summary

GIMPS Discovers 35th Mersenne Prime

O maior ICOSAEDRO do mundo

Paul Erdos morreu dia 20/9/96

(terceiro) maior número primo conhecido é 2^1257787-1

2^{13,466,917}-1 is now the Largest Known Prime December 6, 2001 --> Michael Cameron, a 20 year-old volunteer in a worldwide research project called the Great Internet Mersenne Prime Search (GIMPS), has discovered the largest known prime number using his PC and software by George Woltman and Entropia, Inc. as part of an international grid of more than 205,000 interconnected computers operated by the company. The new number, expressed in shorthand as 2^{13,466,917}-1, contains 4,053,946 digits and was discovered November 14th. It belongs to a special class of rare prime numbers called Mersenne primes . The discovery marks only the 39th known Mersenne prime, named after Marin Mersenne , a 17th century French monk who studied the numbers. Mersenne primes are most relevant to number theory, but most participants join GIMPS simply for the fun of having a role in real research - and the chance of finding a new Mersenne prime. Cameron used a 800 MHz AMD T-Bird PC running part-time for 45 days to prove the number prime. He said, "A friend informed me that if I was going to leave my computer on all the time I should make use of that wasted CPU time. I put GIMPS on my PC because it does not interfere with my work on the computer. Finding the new prime was a wonderful surprise!"

Clay Mathematics Institute Millennium Prize Problems P versus NP The Hodge Conjecture The Poincaré Conjecture The Riemann Hypothesis Yang-Mills Existence and Mass Gap Navier-Stokes Existence and Smoothness The Birch and Swinnerton-Dyer Conjecture In order to celebrate mathematics in the new millennium, The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) has named seven ³Millennium Prize Problems.² The Scientific Advisory Board of CMI selected these problems, focusing on important classic questions that have resisted solution over the years. The Board of Directors of CMI have designated a $7 million prize fund for the solution to these problems, with $1 million allocated to each. During the Millennium meeting held on May 24, 2000 at the Collège de France, Timothy Gowers presented a lecture entitled ³The Importance of Mathematics,² aimed for the general public, while John Tate and Michael Atiyah spoke on the problems. The CMI invited specialists to formulate each problem. One hundred years earlier, on August 8, 1900, David Hilbert delivered his famous lecture about open mathematical problems at the second International Congress of Mathematicians in Paris. This influenced our decision to announce the millennium problems as the central theme of a Paris meeting. The rules that follow for the award of the prize have the endorsement of the CMI Scientific Advisory Board and the approval of the Directors. The members of these boards have the responsibility to preserve the nature, the integrity, and the spirit of this prize. Scientific Advisory Board: Alain Connes Arthur Jaffe Andrew Wiles Edward Witten Directors: Finn M. W. Caspersen Landon T. Clay Lavinia D. Clay William R. Hearst, III Arthur M. Jaffe David B. Stone Paris, May 24, 2000

Faber and Faber today issues a $1,000,000 challenge to prove Goldbach's Conjecture Goldbach's Conjecture was first stated in 1742 in a letter written by Christian Goldbach to the great Swiss mathematician Leonard Euler. The Conjecture is popularly represented as the conjecture that Every even number greater than two is the sum of two primes Although Euler spent much time trying to prove it, he never succeeded. For the next 250 years, other mathematicians would struggle in similar fashion. The proof has not been found to this day, and Goldbach's Conjecture is acknowledged to be one of the most notoriously difficult problems in all of mathematics. On 20 March 2000, Faber and Faber are publishing Uncle Petros and Goldbach's Conjecture, the wonderful and already acclaimed novel by Apostolos Doxiadis. It has been described by John Nash, Nobel Prize Winner as 'a fascinating picture of how a mathematician could fall into a mental trap by devoting his efforts to a too difficult problem' and by George Steiner as 'deeply generous. It allows the lay-reader lucid access to intrinsically closed worlds.' To celebrate publication, we are offering a prize of $1million to any person who can prove Goldbach's Conjecture within the next two years* This challenge is issued in conjunction with Bloomsbury Publishing, USA, the book's American publisher. For further information on the publicity concerning the challenge, please call Judith Hillmore on 0171 465 7554 or e-mail her at judith.hillmore@faber.co.uk Details of how to enter are available with the Rules of the Challenge, or on the Faber and Faber website. *In the event that no satisfactory proof of Goldbach's Conjecture is offered in accordance with the Rules of the Challenge, the reward will not be awarded. No book purchase required.

Research Announcement, February 25, 2000 Proof of the Double Bubble Conjecture by Michael Hutchings, Frank Morgan, Manuel Ritore, and Antonio RosHistory. Archimedes and Zenodorus (see [K, p. 273]) claimed and Schwarz [S] proved that the round sphere is the least-perimeter way to enclose a given volume in R3. The Double Bubble Conjecture, long assumed true (see [P, pp. 300-301], [B, p. 120]) but only recently stated as a conjecture [F1, sect. 3], says that the familiar double soap bubble on the right in Figure 1, consisting of two spherical caps separated by a spherical cap or flat disc, meeting at 120-degree angles, provides the least-perimeter way to enclose and separate two given volumes. The analogous result in R2 was proved by the 1990 Williams College "SMALL" undergraduate research Geometry Group [F2]. In 1995, Hass, Hutchings, and Schlafly [HHS] announced a computer-assisted proof for the case of equal volumes in R3. (See [M1], [HS1], [HS2], [Hu], [M2, chapt. 13].) Here we announce a proof [HMRR] of the general Double Bubble Conjecture in R3, using stability arguments.See also: Frank Morgan: Double Bubble Conjecture Proved

Robert J. Harley's Group Solves ECC2K-95 Exercise Solution Found After 21 trillion elliptic curve operations Seven months after its initial launch, Certicom's ECC Challenge continues to draw attention and major computing efforts from around the world. Certicom congratulates the multiple teams, led by Robert J. Harley of the Institut National de Recherche en Informatique et Automatique (INRIA), France, on their success in solving the Challenge's ECC2K-95 Exercise on May 21, 1998. The solution was found after 21.6 trillion elliptic curve operations, carried out in 25 days by 47 people. Most of the work was done on 200 Alpha machines. Certicom's ECC Challenge is providing cryptographers and mathematicians who are new to the Elliptic Curve Cryptosystem (ECC) the benefit of hands-on experience with elliptic curve algebra. It also serves to increase the industry's understanding and appreciation for the difficulty of the elliptic curve discrete logarithm problem. May 21, 1998 Go to Certicom Go to Status of the Certicom ECC Challenge

"All Syllogistic Arithmetized 'a la Leibniz" Leibniz's 333-year-old problem solved by the Bulgarian logician Vladimir Sotirov. Papers (in DVI and PS formats) and abstracts are available from the: Home Page of Vladimir Sotirov THEOREM 1 (Adequacy of Both Arithmetical Interpretation of the Traditional Syllogistic): A syllogism is true iff it is arithmetically true in the Scholastic as well as in the Leibnizian sense. THEOREM 2 (Adequacy of Both Arithmetical Interpretations of the Syllogistic with Term Negation): A syllogism is true iff it is arithmetically true in the Scholastic as well as in the Leibnizian sense. THEOREM 3 (Adequacy of Both Arithmetical Interpretations of Syllogistic with All Boolean Term Operations): A syllogism is true iff it is arithmetically true in the Scholastic as well as in the Leibnizian sense.

Media Contact: Vredy Lytsman CHRISTIE'S (212) 546-1112 PRESS RELEASE THE ARCHIMEDES PALIMPSEST On October 29 Christie's New York will offer a rare Greek palimpsest manuscript. The earliest witness to the works of Archimedes, the manuscript is the only source for his treatiseOn the Method of Mechanical Theoremsand the only known copy of the original Greek text of his workOn Floating BodiesIt also contains the texts of his worksOn the Measurement of the Circle,On the Sphere and the Cylinder,On Spiral LinesandOn the Equilibrium of Planes. The 174-leaf codex is the most significant and substantial Greek palimpsest known, amd one of the most important palimpsest manuscripts of a classical text ever to come to light. It is estimated to sell for $800,000 - $1,200,000. Archimedes, who lived and worked in Sicily in the 3rd century BC, was the greatest scientist of Antiquity. His most important discoveries were in physics, engineering, optics and astronomy. In these areas he is credited with a number of inventions, among them a planetarium that demonstrated the movement of the stars and planets, the compound pully, the endless screw, a device for raising water, and a variety of military machines used during the defense of Syracuse against the Roman army in 214-212 BC, a siege in which Archimedes himself perished. He also wrote a number of mathematical treatises, elucidating the geometrical properties of complex figures such as spheres, cylinders, circles, spirals and conic sections, and analysing the principles of statics and hydrostatics. His method of proof by exhaustion, as well as his use of infinitesimals, anticipated the development of the calculus, and his works influenced later scientists and mathematician such as Leonardo da Vinci, Gqalileo and Kepler. According to ancient tradition, Archimedes expressed his understanding of the principles of balance and leverage in the remark "Give me a place to stand on, and I will move the earth." He is most populary known for the story associated with the discovery pof specific gravity, when he is said to have run naked through the streets of Syracuse shouting "Eureka!" after he had observed the displacement of water by his body in the bath. The transmission of Archimedes' text depends on only three manuscripts, two of which are now lost. One of these disappeared in the 16th century, after having been copied several times during the Italian Renaissance. The existence of the second was last recorded in the year 1311, after it had served, together with the first manuscript, as a source for William of Moerbeke's medieval Latin translation of a number of the Archimedean treatises. The discovery of the third manuscript, the present codex, at the beginning of the 20th century, was an extraordinary event in classical studies. In addition to offering better readings of four mathematical treatises, it includes the original Greek text ofOn Floating Bodies, previously known only from the medieval Latin translation, and the text of the treatise called theMethod of Mechanical Theorems. The existence of theMethodhad been attested by ancient commentators, but its text had been considered irretrievably lost. Both treatises contribute substantially to understanding Archimedes' work. In theMethod of Mechanical Theorems, Archimedes explained how he used mechanical means to discover the theorems for which he subsequently provided logical mathematical proofs. Not only does this provide valuable insight into the working of the scientist's mind, but the treatise is unique among ancient scientific writings for its treatment of methodology.On Floating Bodiesanalyses the circumstances under which solids of certain shapes will float and also determines the rules of physics governing the displacement of a fluid by solids of varying weights relative to that of the fluid - Archimedes' formal exposition of the principles alluded to in the Eureka story. A palimpsest is a manuscript whose leaves have been written on twice. Although the word comes from the Greek term meaning scraped again, the usual practice was to wash away the first writing after the manuscript was judged to be no longer useful or when there was a shortage of writing materials. In this case, the text of Archimedes, copied in Constantinople in the mid-10th century, was replaced in the 12th with Greek religious texts. The vellum leaves, originally 300 X 200 mm., were also folded in half to make a smaller book, so that the lines of the second script run across those of the first. The importance of palimpsests is that their lower scripts may, as in the case of the Archimedes palimpsest, preserve texts that would otherwise be lost. Well-known examples from classical Latin literature are theRepublicof Cicero and theInstitutesof Gaius, which survive only in palimpsest manuscripts in the Vatican Library and in the Chapter Library at Verona respectively. Other Greek palimpsests, notably of Euripides and Sophocles, preserve fragments that are much shorter, or do not offer significant improvements over the texts of other manuscripts. The existence of the Archimedes palimpsest was first noted in 1899, in a catalogue of the library of the Metochion of the Holy Sepulchre in Istanbul. The Danish scholar J.L. Heiberg examined the manuscript in 1906 and 1908 and incorporated its readings into the second edition of his critical text of Archimedes' works (Bibliotheca Teubneriana, 1910-15). In the 1920s the codex came into the possesion of a French private collector. The other manuscripts from the same Istanbul library are now preserved in institutional libraries in Athens, Paris, Chicago and Cleveland. Since Heiberg saw the Archimedes palimpsest, nearly a century ago, it has been inaccessible to scholarship, and his published transcriptions have been the only record of its readings. Although Heiberg found it possible to decipher substantial passages of the lower script using his naked eye and natural light, the manuscript has never heretofore been systematically examined by means of ultraviolet light or high-resolution digital scanning, the most up-to-date means of recoveriong the readings of palimpsest texts. Recent experiments with the Archimedean palimpsest have shown that the use of ultraviolet light brings up details not easily seen otherwise and that digital images can be manipulated on the computer screen to bring out the lower script while suppressing the upper. The manuscript thus offers an unprecedented opportunity for the marriage of Greek philology and modern technology to further the study of one of the greatest scientists of all time. As the only fundamentally important witness to a classical text not in an institutional library, it also represents the last opportunity for a private collector of scientific books to own an early medieval codex that transmits unique texts from Antiquity. Enquiries: Felix de Marez Oyens on (212) 546 1197 and Hope Mayo on (212) 546 1195 or via fascimile on (212) 980-2043 Auction: Thursday, October 29, 1998, at 2:00 p.m. Location: 502 Park Avenue (59th Street), New York, NY 10022 To order a catalogue: Please call 1-800-395-6300 or 1-718-784-1480 (outside the U.S. and Canada) Photographs are available upon request. Visit Christie's on the Internet: http://www.christies.com

ICM'98 Fields Medalists / Nevanlinna Prize Winner The Fields medals are awarded at each Congress. At ICM 1998 they were awarded to: Richard E. Borcherds (Cambridge University); automorphic forms and mathematical physics. W. Timothy Gowers (Cambridge University); functional analysis and combinatorics. Maxim Kontsevich (IHES Bures-sur-Yvette, and Rutgers Univ.); mathematical physics, algebraic geometry and topology. Curtis T. McMullen (Harvard University); complex dynamics, hyperbolic geometry. The winner of the Nevanlinna prize is Peter W. Shor (AT&T Labs ); quantum computation, computational geometry. Press release Fields Medalists / Nevanlinna Prize Winner Fields

On 2 March 1998 using my PC/Windows program, CP10,Manfred Toplicfound TEN CONSECUTIVE PRIMES IN ARITHMETIC PROGRESSIONThe primes are

P= 100 99697 24697 14247 63778 66555 87969 84032 95093 24689 19004 18036 03417 75890 43417 03348 88215 90672 29719,

P+ 210,P+ 420,P+ 630,P+ 840,P+ 1050,P+ 1260,P+ 1470,P+ 1680 andP+ 1890.For the background of the search see the home page of Tony Forbes. Nik and Michel maintain the results page. Keith Devlin's article in

The Guardian, 19th February 1998. Newsletter/1 : 16 February 1998. Newsletter/2 : 2 March 1998. Newsletter/3 - the official announcement : 7 March 1998.

Congratulations to Roland Clarkson. On January 27th he discovered that 2^3021377 - 1 is prime! This prime number is 909,526 digits long. The computation took 46 days part-time on his 200-MHz Pentium computer. David Slowinski confirmed the find on January 31st. Roland is a 19 year-old sophmore at California State University Dominguez Hills. He is the third youngest Mersenne prime discoverer - behind Noll and Nickel. Incredibly, this was only the 8th exponent he has tested! Unlike the previous GIMPS finds, Roland let the PrimeNet server (see Program News below) choose the lucky exponent. At first, he did not want to test the exponent. Roland said, "I never would have imagined two Mersenne primes would be so close together!". In fact, in percentage terms, the gap between the 36th and 37th Mersenne primes is the smallest ever. To acknowledge Scott Kurowski's work on the PrimeNet server and every GIMPS participants diligent work, official credit for this prime will go to "Clarkson, Woltman, Kurowski, et.al.". You can read the official press release at http://www.mersenne.org/3021377.htm and be sure to check out Chris Caldwell's web pages starting at http://www.utm.edu/research/primes/notes/3021377/

From: Nik LygerosDate: Thu, 22 Jan 1998 11:54:23 +0200 On Nov 7, 1997, we announced on this NMBRTHRY server that we had found 8 consecutive primes in arithmetic progression. We also said that we would try for 9 consecutive primes in AP but would need a lot of help. We got the help, and on Jan 15, 1998, Manfred Toplic Klagenfurt, Austria E-mail: ToplicM@Klagenfurt.Spardat.at informed us that he had just found 9 consecutive primes in Arithmetic Progression. The project was a success! It involved about 100 people using about 200 computers and took about two months. The actual CPU time used was about twice the expected time. We were a little unlucky but we are five very, very happy people. Harvey Dubner Tony Forbes Paul Zimmermann Nik Lygeros Michel Mizony ------------------------------------------------------------------- The solution: 9 consecutive primes in Arithmetic Progression m = 193# =3D product of primes up to 193, m = 19896237639169098164041525154528515360273440272182105821220397609541_ 3910572270. x is the solution for 44 modular equations (see referenced paper), x = 62401416110073076224658890254261851770744681401209443900873273158906_ 59848721. P1 = x + N*m, where N was found after appropriate sieving and testing so that there are 9 consecutive primes in AP, N = 500996388736659, P1 = 9967943206670108648449065369585356163898236408099161839577404858552_ 9071475461114799677694651, P2 = P1 + 210, P3 = P2 + 210, ...., P9 = P8 + 210. We would like to thank Francois Morain for verifying the primality of the 9 primes. We double checked this with the APRT program of UBASIC. --------------------------------------------------------------------- Two programs were used: 1. Tony Forbes wrote the program for PC's running under Windows. 2. Paul Zimmermann wrote the program for work stations and PC's running under Linux. Many thanks to Torbjorn Granlund for making available the free, portable and efficient GMP library, on which the Unix search program was based, and also for suggesting many improvements for that program. We would like to emphasis the contribution of Harry Nelson. Without his idea for generating a "good" x, this project would not have been feasible. Incidently, as a by-product we found 27 new sets of 8 consecutive primes in arithmetic progression. We also found several hundred sets of 7 primes. Reference: H. Dubner, H. Nelson, "Seven Consecutive Primes In Arithmetic Progression," Math. Comp. v66, Oct 1997, pp 1743-1740. ------------------------------------------------------------------------------

------------------------------------------------------------------------------ The November 28 1997 edition of The Chronicle of Higher Education has an article about a banker who discovered a conjecture that is something like Fermat's Last Theorem, and is offering a prize for its proof or disproof. I believe the conjecture, called the Beal Conjecture, after the banker and amateur mathematician that discovered it, is as follows: Given A,B,C,x,y,z whole numbers, x,y,z > 2. If A^x + B^y = C^z, then A, B, C have a common factor. ------------------------------------------------------------------------------

------------------------------------------------------------------------------ | | Date: Thu, 20 Nov 1997 16:35:20 -0600 (CST) | From: Julie Lyden| To: xpolakis@hol.gr | Subject: From Prof. Ribenboim | | | Dear Antreas Hatzipolakis, | | Filip Saidak, from Bratislava (Slovakia) received his B.Sc. from the | University of Auckland (New Zealand). Stimulated by my book on prime | number records, he corresponded with me and showed evidence of a great | talent. I arranged for him to become a graduate student at Queens | University. Since I'm now retired, I could not become his supervisor - | but I keep a close contact with him (when I'm not away). His supervisor | is Ram Murty who will no doubt lead Filip to interesting problems in | number theory. | | On October 23, when I returned from a stay in Rio de Janero, Filip showed | me the "Proof of Carmichael's Conjecture". It used a result attributed | to Grosswald. As I was leaving right away to Urbana-Champaign, I | insisted with Filip that he double check Grosswald's result. As it | turned out: | 1. Filip proved that if n is the smallest exception to Carmichaels | conjecture then 8 does not divide n, a neat and simple proof, but its | result was already in Klee's paper of 1947. | 2. Filip stated that Grosswald proved that the minimal n is divisible by | 32. Unfortunately, he misquoted Grosswald who proved that if phi(n)=a | then 32 divides a, not n. | | It is a pity that Filip-who is only 21- had written to his former | teacher-who is more than 21 and who spread the news without further | verification. | | This little "faux pas" would not have happened ha I been present. Filip, | who is very able, will learn to be more prudent. I'm sure he will have | interesting results in the future. | | Yours, Paulo Ribenboim | | END ------------------------------------------------------------------------- ------------------------------------------------------------------------------

------------------------------------------------------------------------------ FILIP SAIDAK PROVES CARMICHAEL'S CONJECTURE Euler's totient function $\phi(x)$ is defined as the number of natural numbers less than $x$ which are co-prime to $x$. In 1907, R. D. Carmichael conjectured that for each natural number $n$ there exists a natural number $m$ (not equal to $n$) such that $\phi(m)=\phi(n)$. Paulo Ribenboim has surveyed the advances made towards resolving Carmichael's conjecture, in "The Little Book of Big Primes", Springer-Verlag, New York, 1991, pp.25-26. Filip Saidak graduated BSc(Hons) from the University of Auckland in August 1997, and immediately went to Queens University, Kingston, Ontario, for masters study under Paulo Ribenboim. Filip has now proved that famous conjecture - and his proof is short enough to be written on the proverbial postcard! Garry J. Tee (University of Auckland, New Zealand) ------------------------------------------------------------------------------

Date: Tue, 14 Oct 1997 11:31:29 -0400 Sender: Number Theory ListFrom: Harvey Dubner <70372.1170@compuserve.com> Subject: New Amicable Pair record On Oct 4, 1997, Mariano Garcia found an Amicable Pair each of whose members has 4829 digits. This is a new record. The previous largest known pair, found by Frank Zweers in August, has 3766 digits in each member. Zweers and Garcia have discovered new record size Amicable numbers several times in the last ten months. Their findings made use of a theorem contained in the Master's Thesis of Holger Wiethaus. The new pair is the following: N1 = C * M * [(P + Q)*P^89 - 1] N2 = C * Q * [(P - M)*P^89 - 1] C = 2^11 * P^89 M = 287155430510003638403359267 P = 574451143340278962374313859 Q = 136272576607912041393307632916794623 P, Q, (P+Q)*P^89-1 and (P-M)*P^89-1 are prime. This was found using Dubner Crunchers and required about 1 Cruncher-summer of time. The system that found this pair was a 486/66 with a Cruncher. Mariano does not have Email. I will forward any communications to him. Harvey Dubner

A few days ago, Gordon Spencediscovered the largest known prime number using a program written by George Woltman <74473.2626@CompuServe.COM> David Slowinski of Cray Research finished verifying the primality on August 29th. Further information can be found at: http://www.mersenne.org/prime.htm

http://www.utm.edu/research/primes/notes/2976221/

The Erdos Number Project has updated its lists of mathematicians with Erdos number 1 and 2. In the past year, the late Paul Erdos published papers with 10 new people, bringing the total to 472; and his co-authors collaborated with an astonishing 450 people who did not previously have Erdos number less than 3, bringing the total number of people with Erdos number 2 to 5016. Our web site contains lists of these people as well as a wealth of information about Paul Erdos and mathematical collaboration. Its URL is Erdos Numbers Archive. Additions and corrections, or other information that we can add to our pages, should be sent to me. Appended is the README file for the project, containing more details. -- Jerrold W. Grossman, Professor VOICE: (810) 370-3443 Department of Mathematical Sciences FAX: (810) 370-4184 Oakland University FLESH: 322 O'Dowd Hall Rochester, MI 48309-4401 E-MAIL: grossman@oakland.edu WEB: http://www.oakland.edu/~grossman/ =========================================================================

NCTM Response

Executive Summary

=================================================================== Sender: edinfo@inet.ed.gov From: Kirk_Winters@ed.gov (Kirk Winters) Subject: Third International Math & Science Study -- 8th grade X-Comment: Information from & about the U.S. Department of Education (publications & more). INITIAL FINDINGS FROM THE LARGEST, most comprehensive & rigorous international comparison of education ever undertaken were released this week. The report, "Pursuing Excellence," presents initial findings from the Third International Mathematics & Science Study (TIMSS), which examines math & science teaching, learning, curriculum, & achievement across 41 nations. Eighth grade is the focus of this report. It reveals that, among the 41 nations participating in the study, 8th-graders in the U.S. perform above average in science & below average in math. But, Secretary Riley warned at the release of the report (November 20), "If we see the news in the report as simply a horse race story of who finished first & who finished second, we miss the point. The issues are much deeper -- the content & rigor of what we are teaching -- how we go about teaching -- the fact that we continue to shortchange America's teachers by not giving them the preparation & help they need to do the best job possible in the classroom."

========================================================== Executive Summary of "Pursuing Excellence: A Study of U.S. Eighth-Grade Mathematics & Science Teaching, Learning, Curriculum, & Achievement in International Context" ========================================================== Preface ~~~~~~~ The Third International Mathematics & Science Study (TIMSS) is the largest, most comprehensive, & most rigorous international comparison of education ever undertaken. During the 1995 school year, the study tested the math & science knowledge of a half- million students from 41 nations at five different grade levels. In addition to tests & questionnaires, it included a curriculum analysis, videotaped observations of mathematics classrooms, & case studies of policy issues. * TIMSS' rich information allows us not only to compare achievement, but also provides insights into how life in U.S. schools differs from that in other nations. * This report on eighth-grade students is one of a series of reports that will also present findings on student achievement at fourth grade, & at the end of high school, as well as on various other topics. Achievement ~~~~~~~~~~~ One of our national goals is to be "first in the world in mathematics & science achievement by the year 2000," as President Bush & 50 governors declared in 1989. Although we are far from this mark, we are on a par with other major industrialized nations like Canada, England, & Germany. * In mathematics, U.S. eighth graders score below the international average of the 41 TIMSS countries. Our students' scores are not significantly different from those of England & Germany. * In science, U.S. eighth graders score above the international average of 41 TIMSS countries. Our students' scores are not significantly different from those of Canada, England, & Germany. * In mathematics, our eighth-grade students' standing is at about the international average in Algebra; Fractions; and Data Representation, Analysis, & Probability. We do less well in Geometry; Measurement; & Proportionality. * In science, our eighth graders' standing is above the international average in Earth Science, Life Science, & Environmental Issues. Our students score about average in Chemistry & Physics. * If an international talent search were to select the top 10 percent of all students in the 41 TIMSS countries, in mathematics 5 percent of U.S. students would be included. In science 13 percent would be included. Curriculum ~~~~~~~~~~ U.S. policy makers are concerned about whether expectations for our students are high enough, & in particular whether they are as challenging as those of our foreign economic partners. In all countries, the relationship between standards, teaching, & learning is complex. This is even more true in the U.S., which is atypical among TIMSS countries in its lack of a nationally defined curriculum. * The content taught in U.S. eighth-grade mathematics classrooms is at a seventh-grade level in comparison to other countries. * Topic coverage in U.S. eighth-grade mathematics classes is not as focused as in Germany & Japan. * In science, the degree of topic focus in the U.S. eighth-grade curriculum may be similar to that of other countries. * U.S. eighth graders spend more hours per year in math & science classes than German & Japanese students. Teaching ~~~~~~~~ In recent years, concern about the quality of instruction in U.S. classrooms has led mathematics professional organizations to issue calls for reform. However, TIMSS data cannot tell us about the success of these reform efforts for several reasons, including the fact that this assessment occurred too soon after the beginning of the reform for states & districts to have designed their own programs, retrained teachers, & nurtured a generation of students according to the new approach. Also, we do not have comparable earlier baseline information against which to compare the findings from TIMSS. However, TIMSS includes the first large-scale observational study of U.S. teaching ever undertaken, & this can form a baseline against which future progress may be judged. * U.S. mathematics classes require students to engage in less high-level mathematical thought than classes in Germany & Japan. * U.S. mathematics teachers' typical goal is to teach students how to do something, while Japanese teachers' goal is to help them understand mathematical concepts. * Japanese teachers widely practice what the U.S. mathematics reform recommends, while U.S. teachers do so infrequently. * Although most U.S. math teachers report familiarity with reform recommendations, only a few apply the key points in their classrooms. Teachers' Lives ~~~~~~~~~~~~~~~ The training that teachers receive before they enter the profession & the regular opportunities that they have for on-the-job learning & improvement of their teaching affect the quality of the teaching force. The collegial support that teachers receive & the characteristics of their daily lives also affect the type of teaching they provide. * Unlike new U.S. teachers, new Japanese & German teachers undergo long-term structured apprenticeships in their profession. * U.S. teachers have more college education than their colleagues in all but a few TIMSS countries. * Japanese teachers have more opportunities to discuss teaching- related issues than do U.S. teachers. * Student diversity & poor discipline are challenges not only for U.S. teachers, but for German teachers as well. Students' Lives ~~~~~~~~~~~~~~~ The manner in which societies structure the schooling process gives rise to different opportunities & expectations for young people. The motivators, supports, & obstacles to study in each country are outgrowths of the choices provided by society & schools. * Eighth-grade students of different abilities are typically divided into different classrooms in the U.S., & into different schools in Germany. In Japan, no ability grouping is practiced at this grade level. * In mathematics, U.S. students in higher ability-level classes study different material than students in lower-level classes. In Germany & Japan, all students study basically the same material, although in Germany the depth & rigor of study depends on whether the school is for students of higher or lower ability levels. * Japanese eighth-graders are preparing for a high-stakes examination to enter high school at the end of ninth grade. * U.S. teachers assign more homework & spend more class time discussing it than teachers in Germany & Japan. U.S. students report about the same amount of out-of-school math & science study as their Japanese & German counterparts. * Heavy TV watching is as common among U.S. eighth graders as it is among their Japanese counterparts. Conclusions ~~~~~~~~~~~ This report presents initial findings from TIMSS for eighth-grade mathematics & science. A fuller understanding of our nation's educational health must await data from the fourth & twelfth-grade levels. The search for factors associated with student performance is complicated because student achievement after eight years of schooling is the product of many different factors. Furthermore, the U.S. education system is large & decentralized with many interrelated parts. No single factor in isolation from others should be regarded as the answer to improving the performance of U.S. eighth-grade students. With these cautions in mind, this report offers the following insights into factors that may be associated with our students' performance: * The content of U.S. eighth-grade mathematics classes is not as challenging as that of other countries, & topic coverage is not as focused. * Most U.S. mathematics teachers report familiarity with reform recommendations, only a few apply the key points in their classrooms. * Evidence suggests that U.S. teachers do not receive as much practical training & daily support as their German & Japanese colleagues. TIMSS is not an answer book, but a mirror through which we can see our own education system in international perspective. Careful study of our nation's reflection in the mirror of international comparisons will assist educators, business leaders, teachers, & parents as they guide our nation in the pursuit of excellence. =========================================================== To subscribe to (or unsubscribe from) EDInfo, address an email message to: listproc@inet.ed.gov Then write either SUBSCRIBE EDINFO YOURFIRSTNAME YOURLASTNAME in the message, or write UNSUBSCRIBE EDINFO (if you have a signature block, please turn it off). Then send the message. =========================================================== Kirk Winters Office of the Under Secretary U.S. Department of Education kirk_winters@ed.gov ***********************************************************************

How do you slay a giant? One way is to get a very big gun. Most of the previous record primes were found just that way--by the super-computers of their day armed with the Lucas-Lehmer test. Another way is to get together with several hundred friends and assault the giant as a team. Recently the GIMPS (Great Internet Mersenne Prime Search) project did just that! On 13 November 1996 Joel Armengaud discovered the new Mersenne prime 21398269-1. He did this by using George Woltman's free implementation of the Lucas-Lehmer test and by working together with over 700 other individuals scattered across the internet. And what a giant they slew! This 420921 digit number (formed by multiplying two by itself 1398269 times, then subtracting one) is now the largest known prime. It is also the 35th known Mersenne prime, and so comes paired with the 35th known perfect number. There are still infinitely many more giants left to slay, so why not surf over to Woltman's GIMPS site and join the search for the next record prime? ------------------------------------------------------------------------ ORLANDO, Fla., November 23, 1996 -- On November 13, Joel Armengaud discovered the largest known prime number using a program written by George Woltman. Joel Armengaud, a 29-year-old programmer for Apsylog, is from Paris, France. George Woltman is a 39-year-old programmer living in Orlando, Florida. Early this year, Woltman launched the Great Internet Mersenne Prime Search (GIMPS). This web site offers free software for ordinary personal computer owners to use in searching for big prime numbers. Large prime numbers were once the exclusive domain of supercomputer users. "By using a large number of small computers, we negate the supercomputer's speed advantage," said Woltman. Armengaud is one of more than 700 people searching for new primes. Even though Armengaud was the one lucky enough to find this new prime, credit must also go to all the other searchers. Without their efforts, this discovery would not have been possible. The new prime number, 2^1398269-1 is the 35th known Mersenne prime. This prime number is 420,921 digits long. If printed, this prime would fill a 225 page paperback book. It took Joel 88 hours on a 90 MHz Pentium PC to prove this number prime. Armengaud said of his discovery, "Finding this new Mersenne prime was quite a thrill! Mersenne primes are very rare, there was only one chance in 35,000 that this Mersenne number would turn out to be a prime." An integer greater than one is called a prime number if its only divisors are one and itself. For example, the number 10 is not prime because it is divisible by 2 and 5. A Mersenne prime is a prime of the form 2^p-1. The study of Mersenne primes has been central to number theory since they were first discussed by Euclid in 350 BC. The man whose name they now bear, the French monk Marin Mersenne (1588-1648), made a famous conjecture on which values of p would yield a prime. It took 300 years and several important discoveries in mathematics to settle his conjecture. The previous largest known prime, also a Mersenne Prime, was discovered earlier this year using a Cray supercomputer. The new Mersenne prime was independently verified by David Slowinski a co-discoverer of the last Mersenne prime. Although prime numbers of this size have no practical application at the present time, smaller primes have myriad uses including data encryption. Historically the search itself has been more important than the number found. Many results in mathematics were discovered while searching for these primes. The current search has proven useful in at least three respects. First, the search for Mersenne primes is a great way to get youngsters interested in mathematics. One Florida middle school teacher uses the program as a reward. After giving a lesson on prime numbers to her students, anyone who passes the "Prime Quest Test" gets to use a school computer to try and find a new largest known prime. Second, the project has once again proven the power of distributed computing on the Internet. PCs in their spare time can tackle problems that would otherwise require multi-million dollar supercomputers. Said Woltman, "Many other important research projects could use this approach, especially if the funding isn't available for months of supercomputer time. It gives the average person a chance to participate in the scientific discoveries of tomorrow." Finally, the program is a great stress test for PCs. In fact, the program Armengaud used has identified hardware problems in over 3% of the PCs that have run it. Says Woltman, "I'd run the program on any new PC. If it can't pass the self-test, I'd return it." The search for more Mersenne primes is already under way. You can join the Great Internet Mersenne Prime Search at http://ourworld.compuserve.com/homepages/justforfun/prime.htm. You do not need to be a mathematian or computer whiz to use the program. Said Armengaud, "You won't even know the program is running. It uses computer time that would otherwise go to waste." Even though this is the 35th known Mersenne prime, there may be a smaller, as yet undiscovered Mersenne prime. Not all Mersenne numbers between the 32nd and 35th have been checked. There is a well-known formula that generates a "perfect" number from a Mersenne prime. A perfect number is one whose factors add up to the number itself. The smallest perfect number is 6 = 1 + 2 + 3. The 35th known perfect number is 2^1398268 * (2^1398269-1). This number is 841,842 digits long! ------------------------------------------------------------------------Para mais informações ver:

http://www.mersenne.org/prime.htm

http://ourworld.compuserve.com/homepages/justforfun/1398269.htm

http://www.utm.edu/research/primes/notes/1398269.html

## A Student Project at

Lexington High School, Lexington, Massachusetts## by Jason Rosenfeld

(http://forum.swarthmore.edu/students/showcase/largest.icosa/)

We were taking a new class at our school, Modern Geometry. Our teacher, Mr.Kelly, had read on the Net about the icosahedron mentioned in a newsgroup thread in the Math Forum's archives. He gave two groups the assignment to build a larger one. Due to the rivalry between the two groups, our group decided to build a massive 15-foot icosahedron, and set to work.Our group of eight people decided to build the structure out of PVC tubing with vinyl tubing as connector elbows. We would cover the edges with blue tarps to make it a solid. We got the materials, did the construction, and started to put it up, but the tubing bent and it fell down. We tried again, but the wind knocked it around.

We then decided to try building a wooden structure to help support it from the inside. We did this, but when we put the icosahedron on the structure, the wind again blew it down.

We were now very frustrated and in debt, and our group had shrunk to four. We decided to try the structure once more and put it up again.

It stayed up for about 12 hours, but then we got about 8 inches of really heavy snow and most of the tubes and wood bent and were shattered.

Despite already having gotten credit for the project, having the maintenance crew hate us, and being in debt over $300, we decided to rebuild it in the spring using a different approach. Now down to three people, we bought aluminum electrical conduit piping and 480 lb. test shark fishing leader. We drilled holes in the ends of the pipes and put the wire through. We laid the icosahedron out flat on the ground and planned how we were going to put it up.

We gathered two more people and went back to work. After a number of weekends of work we had our icosahedron built--but the wind came back and snapped the fishing line we had used to hold the vertices together.

We got more materials and tried again, this time trying to make the vertices better, and making slits in the tarps. It fell down again. Persistently, we put it up again. This time we used crimping tubes between pipes so the wire couldn't come undone, and we cut off the remaining tarps. This time when we put it up again it only took about five hours. This was in mid-July, and it has been standing 15 feet tall ever since.

We estimate that the whole project took 750 hours.

Now we have moved on to trying to get publicity and documentation forThe Guinness Book of World Records. We have raised over $500 selling candy to pay for it. We have also enlarged our group and have decided to complete the rest of the regular platonic solids in similar sizes. We have already built the tetrahedron and octahedron, and the cube and dodecahedron should be going up in fall of 1996. We have made this a hobby and now do it not for school credit but only pride. We are still selling candy to pay for the new solids and the project has become a complete obsession.The group members on the project, all currently juniors, are: Raphael Bras, Saro Geotzyen, Stephen Hayden, Josh Magri, Nick Varrgelis, David Rosner, and me, Jason Rosenfeld.

The original Lexington High School largest icosahedron Web pages are also available. (Warning: large pictures!)

>Date: Mon, 23 Sep 1996 12:45:50 -0600 >From: juhl@ncsa.uiuc.edu (Jerry Uhl) >Subject: Erdos > >>Date: Sat, 21 Sep 1996 09:31:03 -0500 (CDT) >>From: "John E. Wetzel">> >>Subject: Erdos >> >>I just learned that Paul Erdos died yesterday at a conference >>in Poland, at age 84, active up to the last moment. >> >>--Jack > >Doug West, who was at the confernce, adds: >>Yes, this is true. I had dinner with him the night before. >>There was no hint of any problem. He was in very good spirits, >>repeating his favorite jokes like "A doctor a day keeps the apple away". > >----------------------------------------------------------------------

Form an undirected graph where the vertices are academics, and an edge connects academic X to academic Y if X has written a paper with Y. The Erdos number of X is the length of the shortest path in this graph connecting X with Erdos. Erdos has Erdos number 0. Co-authors of Erdos have Erdos number 1. Einstein has Erdos number 2, since he wrote a paper with Ernst Straus, and Straus wrote many papers with Erdos. The Extended Erdos Number applies to co-authors of Erdos. For People who have authored more than one paper with Erdos, their Erdos number is defined to be 1/# papers-co-authored.

Paul Erdos is an Hungarian mathematician. He obtained his PhD from the University of Manchester and has spent most of his efforts tackling "small" problems and conjectures related to graph theory, combinatorics, geometry and number theory. He is one of the most prolific publishers of papers; and is also and indefatigable traveller. At this time the number of people with Erdos number 2 or less is estimated to be over 4750, according to Professor Jerrold W. Grossman archives. These archives can be consulted via anonymous ftp at vela.acs.oakland.edu under the directory pub/math/erdos. At this time it contains a list of all co-authors of Erdos and their co-authors. On this topic, he writes Let E_1 be the subgraph of the collaboration graph induced by people with Erdos number 1. We found that E_1 has 451 vertices and 1145 edges. Furthermore, these collaborators tended to collaborate a lot, especially among themselves. They have an average of 19 other collaborators (standard deviation 21), and only seven of them collaborated with no one except Erdos. Four of them have over 100 co-authors. If we restrict our attention just to E_1, we still find a lot of joint work. Only 41 of these 451 people have collaborated with no other persons with Erdos number 1 (i.e., there are 41 isolated vertices in E_1), and E_1 has four components with two vertices each. The remaining 402 vertices in E_1induce a connected subgraph. The average vertex degree in E_1 is 5, with a standard deviation of 6; and there are four vertices with degrees of 30 or higher. The largest clique in E_1 has seven vertices, but it should be noted that six of these people and Erdos have a joint seven-author paper. In addition, there are seven maximal 6-cliques, and 61 maximal 5-cliques. In all, 29 vertices in E_1 are involved in cliques of order 5 or larger. Finally, we computed that the diameter of E_1is 11 and its radius is 6. Three quarters of the people with Erdos number 2 have only one co-author with Erdos number 1 (i.e., each such person has a unique path of length 2 to p). However, their mean number of Erdos number 1 co-authors is 1.5, with a standard deviation of 1.1, and the count ranges as high as 13. Folklore has it that most active researchers have a finite, and fairly small, Erdos number. For supporting evidence, we verified that all the Fields and Nevanlinna prize winners during the past three cycles (1986-1994) are indeed in the Erdos component, with Erdos number at most 9. Since this group includes people working in theoretical physics, one can conjecture that most physicists are also in the Erdos component, as are, therefore, most scientists in general. The large number of applications of graph theory to the social sciences might also lead one to suspect that many researchers in other academic areas are included as well. We close with two open questions about C, restricted to mathematicians, that such musings suggest, with no hope that either will ever be answered satisfactorily: What is the diameter of the Erdos component, and what is the order of the second largest component? References Caspar Goffman. And what is your Erdos number? American Mathematical Monthly, v. 76 (1969), p. 791. Tom Odda (alias for Ronald Graham) On Properties of a Well- Known Graph, or, What is Your Ramsey Number? Topics in Graph Theory (New York, 1977), pp. [166-172].

>Largest Known Prime Found by SGI/Cray Supercomputer > >EAGAN, Minn., September 3, 1996 -- Computer scientists at Silicon >Graphics's Cray Research unit, have discovered the largest known prime >number while conducting tests on a CRAY T90 series supercomputer. > >The new prime number has 378,632 digits. Printed in newspaper-sized >type, the number would fill approximately 12 newspaper pages. > >In mathematical notation, the new prime number is expressed as >2^1257787-1 , which denotes two, multiplied by itself 1,257,787 times, >minus one. Numbers expressed in this form are called Mersenne prime >numbers after Father Marin Mersenne, a 17th century French monk who >spent years searching for prime numbers of this type. > >Prime numbers can be divided evenly only by themselves and one. Examples >include 2, 3, 5, 7, 11 and so on. The Greek mathematician Euclid proved >that there are an infinite number of prime numbers. But these numbers do >not occur in a regular sequence and there is no formula for generating >them. Therefore, the discovery of new primes requires randomly >generating and testing millions of numbers. > >Silicon Graphics employees have been at the leading edge for prime >number discoveries since 1978. The last 10 records for the largest known >prime belong to people who are now working at Silicon Graphics: >Prime Digits Year SGI Employee > >2^21701-1 6533 1978 Landon Curt >Noll > (with Laura Nickel) >2^23209-1 6987 1979 Landon Curt >Noll >2^44497-1 13395 1979 David Slowinski > (with Harry >Nelson) >2^86243-1 25962 1982 David Slowinski >2^132049-1 39751 >1983 David Slowinski >2^216091-1 65050 1985 David Slowinski >391591 * > 65087 1989 Landon Curt Noll > 2^216193-1 John Brown > > Sergio Zarantonello > (with Joel & Gene Smith, Bodo Parady) > >2^756839-1 227832 1992 David Slowinski > Paul Gage >2^859433-1 > 258716 1994 David Slowinski > Paul Gage >2^1257787-1 378632 >1996 David Slowinski > Paul Gage >"Finding these special numbers is a true 'needle-in-a-haystack' >exercise, but we improve our odds by using a tremendously fast computer >and a clever program," said David Slowinski, a Cray Research computer >scientist. and fellow Cray Research computer scientist Paul Gage >developed the program that found the new prime number. Mathematician >Richard Crandall (of NeXT), independently verified that the number >Slowinski and Gage found is prime. > >Prime numbers have applications in cryptography and computer systems >security. Huge prime numbers like those discovered most recently are >principally mathematical curiosities, but the process of searching for >prime numbers does have several practical benefits. > >For instance, the "prime finder" program developed by Slowinski and Gage >is used by Silicon Graphics's Cray Research unit as a quality assurance >test on all new supercomputer systems. A core element of this program is >a routine that involves squaring a number repeatedly. As this process >continues, it eventually involves multiplying immense numbers -- numbers >of hundreds of thousands of digits -- by themselves. > >"This acts as a real 'torture test' for a computer," said Slowinski. >"The prime finder program rigorously tests all elements of a system -- >from the logic of the processors, to the memory, the compiler and the >operating and multitasking systems. For high performance systems with >multiple processors, this is an excellent test of the system's ability >to keep track of where all the data is." Slowinski said the recent CRAY >T90 series supercomputer test in which the new prime number was >discovered would run for over 6 hours on one central processing unit of >the system. "If a machine can complete this exhaustive run-through, we >can be confident everything is working as it should," said Slowinski. > >In addition, Slowinski said, techniques used to speed up the performance >of the prime finder can also be used to enhance the performance of >programs customers use on real-world problems such as forecasting the >weather and searching for oil. "Through our work on the prime finder >program, we learn new techniques for speeding up certain kinds of >mathematical operations. These operations are often key elements of the >most computation-intensive portions of software programs our customers >run on their systems," said Slowinski. > >Slowinski compared running the prime finder on supercomputers and >continually "tuning" the program to building and racing exotic cars. >"There aren't many practical uses for dragsters or Formula 1 race cars. >But some things engineers do to make those cars perform better >eventually find their way into cars you and I drive," said Slowinski. > >Slowinski noted that with the discovery of the new prime number, a new >perfect number can also be generated. A perfect number is equal to the >sum of its factors. For example, 6 is perfect because its factors -- 1, >2 and 3 -- when added together, equal 6. Mathematicians don't know how >many perfect numbers exist. They do know, however, that all perfect >numbers have a direct relationship to Mersenne primes. The new perfect >number generated with the new Mersenne prime is the 34th known perfect >number and has 757,263 digits. > >------------------------------------------------------------------------ > > >Mais informacoes na WEB em: > http://www.utm.edu/research/primes/notes/1257787.html http://www.sjmercury.com/business/compute/prime.htm http://reality.sgi.com/csp/ioccc/noll/prime/prime_press.html http://ourworld.compuserve.com/homepages/justforfun/prime.htm http://www.utm.edu/research/primes/ http://www.utm.edu/research/primes/largest.html > > >---------------------------------------------------------------- > >

Última alteração: 8 de Janeiro de 2002