As últimas do mundo da Matemática

Índice:

Novo recorde: 39th Known Mersenne Prime Found!!

213,466,917-1 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 213,466,917-1, 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!"

Millennium Prize Problems

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

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

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.

Double Bubble Conjecture Proved

Research Announcement, February 25, 2000

Proof of the Double Bubble Conjecture
                   

       by Michael Hutchings, Frank Morgan, Manuel Ritore, and Antonio Ros


History. 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

Novo recorde: O grupo GIMPS encontra o seu quarto número primo!!!!

2^{6 972 593}-1 é o 38^o número primo de Mersenne conhecido

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

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

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 

Leibniz's 333-year-old problem solved

"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. 

Palimpsest with On the Method of Mechanical Theorems by Archimedes

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 treatise On the Method of Mechanical Theorems and the
only known copy of the original Greek text of his work On Floating Bodies
It also contains the texts of his works On the Measurement of the Circle,
On the Sphere and the Cylinder, On Spiral Lines and On 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
of On Floating Bodies, previously known only from the medieval Latin
translation, and the text of the treatise called the Method of Mechanical
Theorems. The existence of the Method had been attested by ancient
commentators, but its text had been considered irretrievably lost. Both
treatises contribute substantially to understanding Archimedes' work. In the
Method 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 Bodies
analyses 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 the Republic of Cicero and the
Institutes of 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

Fields Medalists / Nevanlinna Prize 1998

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

ten consecutive primes in arithmetic progression

On 2 March 1998 using my PC/Windows program, CP10, Manfred Toplic

found TEN CONSECUTIVE PRIMES IN ARITHMETIC PROGRESSION

The 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 and P + 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.

37th Known Mersenne Prime Discovered!!!

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/

New Math. Record: primes in arithmetic progression

From: Nik Lygeros 
Date: 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.


------------------------------------------------------------------------------

Beal Conjecture

------------------------------------------------------------------------------

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.

------------------------------------------------------------------------------

Falso alarme: CARMICHAEL'S CONJECTURE

------------------------------------------------------------------------------
|
| 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

------------------------------------------------------------------------------

                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) 

------------------------------------------------------------------------------

New Amicable Pair record

Date:         Tue, 14 Oct 1997 11:31:29 -0400
Sender:       Number Theory List 
From:         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

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

A few days ago, Gordon Spence 
discovered 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/

 

Erdos Numbers update

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/
=========================================================================

Third International Math & Science Study -- 8th grade

===================================================================
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."

Third International Math & Science Study -- 8th grade - Executive Summary

==========================================================
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.

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GIMPS (Great Internet Mersenne Prime Search) Finds a Prime!

2^1398269-1 is prime

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! 
------------------------------------------------------------------------

O maior ICOSAEDRO do mundo

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 for 
The 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!) 

 

Paul Erdos morreu dia 20/9/96

>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".
>
>----------------------------------------------------------------------

Erdos Number

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. 

Who is Paul Erdos?

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]. 

Maior número primo conhecido é 2^1257787-1

>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
>
>
>----------------------------------------------------------------
>
>

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