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         Slowinski David:     more detail
  1. The 32nd Mersenne Prime - Predicted by Mersenne by David Slowinski, 2010-07-06

21. LitSearch: An Online Literary Database
Slowinski, David (00) Works by this author 32nd Mersenne Prime, The; predictedby Mersenne. Copyright 2001 Keith Ito. All Rights Reserved. Admin Control Panel.
http://daily.stanford.edu/litsearch/servlet/DescribeAuthor?name=Slowinski, David

22. LitSearch: An Online Literary Database
Simms, W. Gilmore (William Gilmore) Sinclair, Upton Skelton, Oscar Douglas Skinner,Constance Lindsay Slaveikov, Pencho P. Slowinski, David Smiles, Samuel Smith
http://daily.stanford.edu/litsearch/servlet/DescribeAuthor?name=S

23. Large Prime Numbers
1978 Landon Curt Noll (with Laura Nickel, now Ariel Glenn) 2^232091 6987 1979 LandonCurt Noll 2^44497-1 13395 1979 David Slowinski (with Harry Nelson) 2^86243
http://www.isthe.com/chongo/tech/math/prime/prime_press.html
Large Prime Number Found by SGI/Cray Supercomputer
Now serving
text by: Landon Curt Noll
Note: This is no longer the largest known prime
The largest known prime number may be found in chongo's table of
Mersenne Prime Digits and Names
<== try me EAGAN, Minn., September 3, 1996 Computer scientists at SGI 's former Cray Research unit, have discovered a large prime number while conducting tests on a CRAY T90 series supercomputer. The 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 , which denotes two, multiplied by itself 1,257,787 times, minus one. Numbers expressed in this form are called Mersenne prime numbers after Marin Mersenne, a 17th century French monk who spent years searching for prime numbers of this type. See Chris Callwell's prime page for more information on prime numbers. Prime numbers can be divided evenly only by themselves and one. Examples include 2, 3, 5, 7, 11 and so on. The Greek mathematician

24. SAVEgateway Document Delivery
systems that in turn help solve realworld problems such as cryptography, improvingweather forecasts and designing safer cars, said David Slowinski, the other
http://www.isthe.com/chongo/tech/math/prime/mercnews.html
Note: This is no longer the largest known prime
The largest known prime number may be found in chongo's table of Mersenne Prime Digits and Names
CRUNCHING NUMBERS
RESEARCHERS MAKE PRIME MATH DISCOVERY
Published: Tuesday, September 3, 1996
Section: Front
Page: 1A
BY DAN GILLMOR, Mercury News Computing Editor
When the British mountaineer George Leigh Mallory was asked why he wanted to scale Mount Everest, he replied: ''Because it's there.'' A related urge sparks computer scientists at Silicon Graphics Inc.'s Cray Research unit, who will announce today that they've discovered the world's largest-known prime number - and a special kind of prime number at that. This one is 378,632 digits long, roughly 120 single-spaced typewritten pages - and ''a rare jewel,'' said co-discoverer Paul Gage. But the way they found it, using sophisticated programming on high-powered supercomputers, goes well beyond mathematical mountain climbing. The techniques help create and test computer systems that in turn help solve real-world problems such as cryptography, improving weather forecasts and designing safer cars, said David Slowinski, the other co-discoverer of the latest record number. Using a Cray T94 supercomputer, Slowinski and Gage found what is currently the biggest example of a Mersenne prime number, named after a 17th-century French monk, Father Marin Mersenne, who had a thing for numbers. A prime number is an integer greater than zero whose divisors are only itself and 1. (The number 2 is prime because it can only be divided evenly by 1 and 2, for example). Mersenne numbers are primes that take the form 2 to some power, minus 1 - in other words, 2 multiplied by itself a certain number of times with 1 subtracted from the result.

25. Lettuce.edsc.ulst.ac.uk/gimps/programs.txt
D1,9,David Slowinski D2,10,David Slowinski D3,11,David Slowinski D4,12,David SlowinskiD5,13,David Slowinski D6,14,David Slowinski D7,15,David Slowinski D8,16
http://lettuce.edsc.ulst.ac.uk/gimps/programs.txt
A1,28,Amdahl 6 - 1989 A2,30,Amdahl 6 - 1989 B1,64,Dave Boyd C1,64,Nick Craig-Wood. StongARM machines. D1,9,David Slowinski D2,10,David Slowinski D3,11,David Slowinski D4,12,David Slowinski D5,13,David Slowinski D6,14,David Slowinski D7,15,David Slowinski D8,16,David Slowinski D9,17,David Slowinski E,64,Ernst Mayer v2.3 through 2.4c - residues may have up to 3 digits wrong E1,64,Ernst Mayer v2.4d - residue bug fixed. E2,64,Ernst Mayer v2.5 G,15,Gary Gostin G2,64,Glucas v. 2.2 G29,64,Glucas with initial shift counts J,12,John Sweeney. Mac version 1.1 - bug in residue code. J1,64,John Sweeney. Mac version 1.2 - same as 1.1 but residue bug fixed. J2,64,John Sweeney. Mac version 1.3 - radix 4 implementation J3,64,John Sweeney. Mac version 1.4 - Bug fix. prev versions can give bad data J4,64,John Sweeney. MacLL v1.0b1 M,64,Peter Marksteiner. M1,64,Woltman - OS/2 Version of WPn (Marcel van de Vusse) M3,64,Woltman - OS/2 Version of WQn (Michel van Loon) N1,15,Nick Myrman. Home grown FFT. residues up to 15 bits - can be off by 1. N2,15,Nick Myrman. Crandall FFT. residues up to 15 bits - can be off by 1. O1,64,Woltman - OS/2 version of WPn (Matan Ziv-Av) S,15,Dave Smitley U,15,Unknown U2,28,Unknown (sent to David Slowinski) W1,0,Woltman - All integer version W2,64,Woltman - Early floating point version W4,64,Woltman - First Web release W5,64,Woltman - Separate lucas14 and 15. pre-factoring - Win 3.1 W6,64,Woltman - Windows 95 version of W5 - Win 95 W7,64,Woltman - Vastly improved factoring algorithm - Win 3.1 W8,64,Woltman - Vastly improved factoring algorithm - Win 95 W9,64,Woltman - Better self-test. fixed factoring continue bug - Win 3.1 WA,64,Woltman - Better self-test. fixed factoring continue bug - Win 95 WB,64,Woltman - No two-to-phi array. more error checking - Win 3.1 WC,64,Woltman - No two-to-phi array. more error checking - Win 95 WL,64,Woltman - No two-to-phi array. more error checking - Linux WP0,64,Woltman - PFA version - Win 3.1 WP1,64,Woltman - PFA version - Win 95 WP2,64,Woltman - PFA version - Linux WP3,64,Woltman - PFA version - Win 3.1 Screen Saver WP4,64,Woltman - PFA version - Win 95 Screen Saver WP5,64,Woltman - PFA version - Win NT Service WQ0,64,Woltman - Better error checking - Win 3.1 WQ1,64,Woltman - Better error checking - Win 95 WQ2,64,Woltman - Better error checking - Linux WQ3,64,Woltman - Better error checking - Win 3.1 Screen Saver WQ4,64,Woltman - Better error checking - Win 95 Screen Saver WQ5,64,Woltman - Better error checking - Win NT Service WQ6,64,Woltman - Better error checking - UNIXWare version compiled by MF WR1,64,Woltman - Networked version - Win 95 WR2,64,Woltman - Networked version - Linux WR5,64,Woltman - Networked version - Win NT Service WR7,64,Woltman - Networked version - OS/2 port by Michiel van Loon WS0,64,Woltman - Exp to 20.5M - Win 3.1 WS1,64,Woltman - Exp to 20.5M - Win 95 WS2,64,Woltman - Exp to 20.5M - Linux WS5,64,Woltman - Exp to 20.5M - Win NT Service WS7,64,Woltman - Exp to 20.5M - OS/2 port by Michiel van Loon WT0,64,Woltman - Shifted starting value - Win 3.1 WT1,64,Woltman - Shifted starting value - Win 95 WT2,64,Woltman - Shifted starting value - Linux WT5,64,Woltman - Shifted starting value - Win NT Service WT7,64,Woltman - Shifted starting value - OS/2 port by Michiel van Loon WU0,64,Woltman - Works above 2^22 - Win 3.1 WU1,64,Woltman - Works above 2^22 - Win 95 WU2,64,Woltman - Works above 2^22 - Linux WU5,64,Woltman - Works above 2^22 - Win NT Service WU7,64,Woltman - Works above 2^22 - OS/2 port by Michiel van Loon WV1,64,Woltman - Even faster! - Win 95 WV2,64,Woltman - Even faster! - Linux WV3,64,Woltman - Even faster! - Solaris WV5,64,Woltman - Even faster! - Win NT Service WV6,64,Woltman - Even faster! - FreeBSD WV7,64,Woltman - Even faster! - OS/2 port by Michiel van Loon WW1,64,Woltman - P-1 - Win 95 WW2,64,Woltman - P-1 - Linux WW5,64,Woltman - P-1 - Win NT Service WW6,64,Woltman - P-1 - FreeBSD WX1,64,Woltman - SSE2 and prefetch - Win 95 WX2,64,Woltman - SSE2 and prefetch - Linux WX5,64,Woltman - SSE2 and prefetch - Win NT Service WY1,64,Woltman - New FFT crossovers WY2,64,Woltman - New FFT crossovers WY5,64,Woltman - New FFT crossovers X,64,Richard Crandall program and it's successors - UNIX X1,32,Crandall's program - UNIX

26. Mersenne Primes
Curt Noll Laura A. Nickel (Cyber 174) 26 23209 6987 13973 1979 Landon Curt Noll(Cyber 174) 27 44497 13395 26790 1979 David Slowinski Harry L. Nelson (Cray
http://pw1.netcom.com/~hjsmith/Perfect/Mersenne.html
Mersenne Primes
N is an even perfect number if and only if
N = 2^(q-1) * (2^q - 1) and 2^q - 1 is prime
It should also be noted that for 2^q - 1 to be prime q must be prime. So when we search for even perfect numbers, we search on q equal to the primes. The numbers M(q) = 2^q - 1 (with q prime) are called Mersenne numbers. If M(q) = is prime then it is called a Mersenne prime. If a prime q makes a Mersenne number a Mersenne prime, then P(q) = 2^(q-1) * (2^q - 1) is a Perfect number. Here are the 39 known Mersenne primes, M(q), as of Nov 14, 2001:
The way to determine if 2^q - 1 is prime, given that q is an odd prime, is to use the Lucas-Lehmer test:
Lucas-Lehmer-Test(q): u := 4 for i := 3 to q do u := (u^2 - 2) mod (2^q - 1) enddo if u == then 2^q - 1 is prime else 2^q - 1 is composite endif EndTest
Return to Perfect Numbers
Return to Harry's Home Page
This page accessed times since June 7, 1998.

27. Cruise Chip
27, 44,497, 8.54509 *10^13394, 1979, Harry L. Nelson David Slowinski, Cray1.28, 86,243, 5.36927 *10^25961, 1982, David Slowinski Lawrence Livermore Lab,
http://www.cruisechip.com/mersenne.htm
Home Resume Mathematics Loads Links Site Map Cruise Chip is the combination of the new technology of intelligence encapsulation and the classical metatheme of moneyed luxury. Enjoy the paradigm! A Historical Summary of Mersenne Primes This is a copy of similar tables brought up to date and merged together. Please E-mail me any corrections. Note for example that M10 is given as 6.18970 *10^26. This represents the 27-digit number 618,970,019,642,690,137,449,562,111. Mersenne Prime Table M Exponent
Mersenne
prime p Proved
Prime Proven
by Machine
Used Antiquity Euclid's Elements Codex Lat Monac., 14808 Pietro Antonio Cataldi
Leonhard Euler I.M. Pervouchine
Seelhof R. E. Powers R. E. Powers
E. Fauquembergue
Eduard Lucas
E. Fauquembergue Raphael M. Robinson Natl. Standard Bureau SWAC Raphael M. Robinson Lehmer Lehmer Lehmer Hans Riesel BESK Sep 1961 Alexander Hurwitz Selfridge IBM-7090 Alexander Hurwitz Selfridge Donald B. Gillies

28. Bookshare.org - Books By Author
Please log in. Books by David Slowinski. Here is a list of our books by David Slowinski .There is 1 book by this author in our collection. This is book 1 of 1.
http://www.bookshare.org/web/BooksByAuthor.html?author_id=180

29. Bookshare.org - Books By Author
John Sladek. Frank G. Slaughter. Jim Sleeper. E. Dendy Sloan. Joan Slonczewski.David Slowinski. Lass Small. Julie Smart. Samuel Smiles. Jane Smiley. Adam Smith.
http://www.bookshare.org/web/BooksByAuthor.html?authorstring=S&firstlast=N

30. Nanobiographies
Adi ; Shanks, Daniel Charles (1917) ; Slowinski, David ; Spiro, Claudia
http://algo.inria.fr/banderier/Recipro/node53.html
Next: Remerciements Up: Previous:
Nanobiographies
On devrait trouver ci-dessous les noms des intervenants plus ou moins illustres de ce T.E.R.
Bernoulli, Jacques I er (1654-1705) suisse
Cassels, John William Scott (1922) anglais
Dedekind, Richard Julius Wihelm (1831-1916) allemand
Diophante d'Alexandrie (III e
Dirichlet, Peter Gustav Lejeune- (1805-1859) allemand
Eisenstein, Ferdinand Gotthold (1823-1852) allemand
Euclide (III e
Euler, Leonhard (1707-1783) allemand
Fields, John Charles (1863-1932) canadien
Gauss, Karl Friedrich (1777-1855) allemand Goldbach, Hermann (1690-1764) allemand Hardy, Godfrey Harold (1877-1947) anglais Hasse, Helmut (1898-1980) allemand Hecke, Erich (1887-1947) polonais Hensel, Kurt (1861-1941) allemand Hilbert, David (1862-1943) allemand Jacobi, Carl Gustav (1804-1851) allemand Jensen, Johannes Ludwig Wilhelm (1859-1925) allemand Kronecker, Leopold (1823-1891) allemand Kummer, Ernst Eduard (1810-1893) allemand Leibniz, Gottfried William (1646-1716) allemand Linnik, Iurii Vladimirovich (1915-1972) russe Ostrowski, Alexander Markus (1893)

31. Maiores_primos
David Slowinski,1985. 2 132049 -1, 39751, David Slowinski, 1983. 2 86243 -1, 25962, David Slowinski,1982.
http://www.educ.fc.ul.pt/icm/icm98/icm12/recordistas.htm
ICM do DEFCUL
[Os maiores, dos maiores,

primos existentes]

primos de Mersenne]
...
primos de Sophie Germain]
Os maiores, dos maiores, primos existentes
Mersenne PrimeNet Great Internet Mersenne Prime Search
Quem Quando [Voltar ao topo]
Os maiores
Quem Quando Harvey Dubner Tony Forbes Harvey Dubner

[Voltar ao topo]

Mersenne
Mersenne p
Quem Quando David Slowinski David Slowinski David Slowinski [Voltar ao topo]
Os maiores primos primodiais e factoriais
n primos primordiais n
Quem Quando Charles F. Kerchner III Charles F. Kerchner III Chris Caldwell Chris Caldwell Chris Caldwell Chris Caldwell Charles F. Kerchner III Harvey Dubner
[Voltar ao topo]
primos de Sophie Germain
primo de Sophie Germain
Quem Quando Yves Gallot Harvey Dubner Harvey Dubner Harvey Dubner Harvey Dubner
[Voltar ao topo]

32. PRIS Meeting Minutes, 2 December 1996
ATTENDANCE Board John d'Amato, Ed Green, Jerry Kennedy, Bill Kusk, Edith Leer,Chad McDowell, Gordon Moffatt, John Slowinski, David Trudeau RegretsLarry
http://www.pris.bc.ca/pris/prisinfo/minutes/dec2_96.html
Board of Directors Meeting: 2 December 1996
Technology Center, Dawson Creek
ATTENDANCE:
Board:
John d'Amato, Ed Green, Jerry Kennedy, Bill Kusk, Edith Leer, Chad McDowell, Gordon Moffatt, John Slowinski, David Trudeau
Regrets: Larry Legault, Rick Saunders
Guests: Arvo Koppel, Brian McNeil Meeting called to order at 7:40 pm
M/S/C to adopt agenda
M/S/C to accept minutes of 4 Nov, 96 (Slowinski, Kusk) BUSINESS ARISING:
Endorsed motions of meeting of 4 Nov 96 were subsequently carried by e-mail vote.
Motion of 10 Nov to increase B. McNeil's wage was carried by e-mail vote.
Appropriate notifications and press releases for Library access, and Ag assistance were issued. Committee to review staffing requirements met, and job descriptions were developed, indicating a need for three full-time staff positions. Job descriptions are published as "Duties of PRIS Staff" SYSOP REPORT Membership Numbers System Configuration System Expansion Requirements FINANCIAL REPORT SUMMARY as at 1 DECEMBER 1996 Bank Balances Accounts Receivable, Other than Memberships

33. PRIS Meeting Minutes, 4 November 1996
ATTENDANCE Board John d'Amato, Bill Kusk, Edith Leer, John Slowinski, David TrudeauRegrets Ed Green, Jerry Kennedy, Larry Legault, Chad McDowell, Gordon
http://www.pris.bc.ca/pris/prisinfo/minutes/nov4_96.html
Board of Directors Meeting: 4 November 1996
Technology Center, Dawson Creek
ATTENDANCE:
Board:
John d'Amato, Bill Kusk, Edith Leer, John Slowinski, David Trudeau
Regrets: Ed Green, Jerry Kennedy, Larry Legault, Chad McDowell, Gordon Moffat, Rick Saunders
Guests: Arvo Koppel (Sysop) Meeting called to order at 7:38 pm, with no quorum
M/S/C to adopt agenda
Minutes to October meeting to be available at later date. BUSINESS ARISING:
/NA SYSOP REPORT
Membership Numbers

System Configuration
System Expansion Requirements FINANCIAL REPORT SUMMARY as at 1 NOVEMBER 1996 Bank Balances Accounts Receivable, Other than Memberships Accounts Receivable, Memberships Credit on Membership Accounts Prepayments for Future Services Accounts Payable Liabilities LIQUID POSITION Capital Assets Less Accumulated Depreciation BALANCE UPDATE ON CONTRACT WITH SD 59 John d'Amato reports that the School Board is unable to sign a contract with PRIS for the exchange of Internet services for building rental under the previously negotiated terms, because of a provincial initiative to provide all schools, libraries, colleg es, and museums with internet access under the PLNet initiative. PRIS has been assured verbally by local school board officials that the existing arrangement with PRIS is not threatened in principle. School District 59 has opened a corporate account so that all school board employees can obtain PRIS Internet accounts under the corporate rate structure, for dial-in access. All member support, and all billing will be handled by the school district, und er a payroll deduction plan.

34. Www.mersenne.org/news1.txt
David Slowinski As most of you know, David Slowinski has beensearching for Mersenne primes for 17 years using spare CPU cycles on his
http://www.mersenne.org/news1.txt

35. 35th Mersenne Prime Discovered
supercomputer. The new Mersenne prime was independently verifiedby David Slowinski a codiscoverer of the last Mersenne prime.
http://www.mersenne.org/1398269.htm

36. Sciaga.pl - Liczby Pierwsze
ma przede wszystkim wchodzaca w sklad specjalnego zespolu Silicon Graphic’sCray Research slawna para matematyków David Slowinski – Paul Gage.
http://slimak.sciaga.pl/prace/praca/5640.htm
sciaga prace / przedmiot: Matematyka reklama kontakt info Uwaga - tylko u Nas! masz mo¿liwo¶æ sci±gniêcia nowych gier do telefonów komórkowych napisanych w JAVIE. Je¿eli posiadasz telefon marki NOKIA (3510i, 8910i, 3530, 3560, 7560, 3410, 6310i, 3650) lub SIEMENS (MT50, M50, C55) to skorzystaj ju¿ dzi¶. zobacz wiêcej Temat: Liczby Pierwsze
Liczby pierwsze s± to takie liczby naturalne, które wiêksze s± od jedynki i podzielne bez reszty przez sam± siebie i jedynkê. Jednym z pytañ dotycz±cych liczb pierwszych, które narzuca siê ka¿demu jest pytanie o liczbê tych liczb: ile ich jest, skoñczenie wiele czy, wrêcz przeciwnie, nieskoñczenie? Na to akurat znamy odpowied¼ od czasów staro¿ytnych: liczb pierwszych jest nieskoñczenie wiele. Wiedzia³ o tym ju¿ w IV w. p.n.e. sam wielki Euklides. Innymi s³owy, nie istnieje najwiêksza liczba pierwsza: dla ka¿dej danej liczby pierwszej mo¿emy znale¼æ wiêksz±. Istotnie, gdyby by³a jedynie skoñczona liczba liczb pierwszych (np. P) to iloczyn wszystkich tych P liczb, zwiêkszony o jedynkê, musia³by byæ te¿ pierwszy (bo przy dzieleniu przez któr±kolwiek z tych P liczb dawa³by oczywi¶cie. resztê jeden); zatem przypuszczenie, ¿e jest ich P, jest fa³szywe, bowiem znale¼li¶my oto nastêpn±.
Ta do¶æ prosta konstrukcja daje równie¿ teoretycznie przepis na konstruowanie coraz wiêkszych liczb. Np. 2*3*5+1=31; 31 jest liczb± pierwsz±. Nietrudno spostrzec , ¿e ten przepis ma te¿ wady. Nie mo¿na nim otrzymaæ np.: 11,13,17,19,...); po drugie omówiony powy¿ej zapis konstruowanej liczby trudno uznaæ za przejrzysty. Dodam jeszcze, ¿e w ogóle nie istnieje - i nie mo¿e istnieæ - ¿aden "wzór", w zwyk³ym sensie tego s³owa który by "produkowa³" wszystkie liczby po kolei. Nie oznacza to, i¿ nie mo¿na podaæ wzorów, które daj± nam ca³± seriê takich liczb, dowolnej zreszt± d³ugo¶ci. Skoro liczb pierwszych jest nieskoñczenie wiele, to mo¿e do odnajdywania kolejnych przyda siê inny algorytm.

37. SFGate
Joseph Slowinski SF biologist, expert on snakes David Perlman, Chronicle ScienceEditor Wednesday, September 19, 2001 ©2002 San Francisco Chronicle.
http://jacq.org/jbs-sfgate.htm
Back
www.sfgate.com
Return to regular view Joseph Slowinski S.F. biologist, expert on snakes
David Perlman, Chronicle Science Editor
Wednesday, September 19, 2001
©2002 San Francisco Chronicle
URL: http://www.sfgate.com/cgi-bin/article.cgi?file=/chronicle/archive/2001/09/19/MN192810.DTL Joseph B. Slowinski, a noted San Francisco biologist and one of the world's leading experts on venomous snakes, died from a paralyzing snakebite on Sept. 12 while leading an expedition in the jungles of northern Burma. He was 38. As associate curator of herpetology at the California Academy of Sciences, Dr. Slowinski was known as a bold, high-spirited scientist and a brilliant biologist whose studies of the evolutionary history of the cobra family have proved uniquely valuable to science. His fatal encounter with a krait, a member of the cobra group, occurred when a member of his Burmese team brought him a sack containing a single small reptile whose coloration resembled a familiar harmless snake. Dr. Slowinski reached into the sack, and the snake bit him painlessly as he grasped, according to an e-mail message from Douglas J. Long, the acting chairman of the academy's ornithology department, who was collecting bird specimens on the expedition.

38. The Largest Known Primes
The primality of this number was verified by David Slowinski who has found severalof the recent record primes. 2 216091 1, 65050, David Slowinski, 1985,
http://w3.impa.br/~gugu/mersenne/largest.html
The Largest Known Primes
Contents:
  • Introduction (What are primes? Who cares?)
  • The Top Ten Record Primes
    largest
    twin ... Mersenne , and Sophie Germain
  • The Complete List of the Largest Known Primes
  • Other Sources of Prime Information
  • Euclid's Proof of the Infinitude of Primes ...
  • Comments? Suggestions? New records? New Links? Primes: Home Largest Proving How Many? ... Guestbook Note: The correct URL for this page is http://www.utm.edu/research/primes/largest.html
    1. Introduction
    An integer greater than one is called a prime number if its only positive divisors (factors) are one and itself. For example, the prime divisors of 10 are 2 and 5; and the first six primes are 2, 3, 5, 7, 11 and 13 ( the first 10,000 , and other lists are available). The Fundamental Theorem of Arithmetic shows that the primes are the building blocks of the positive integers: every positive integer is a product of prime numbers in one and only one way, except for the order of the factors. The ancient Greeks proved (ca 300 BC) that there were infinitely many primes and that they were irregularly spaced (there can be arbitrarily large gaps between successive primes ). On the other hand, in the nineteenth century it was shown that the number of primes less than or equal to
  • 39. ÒµÓàÊýѧÌìµØ-ÊýÂÛ֪ʶ-×î´óµÄËØÊý
    1992, Mersenne 32. 31, 2 216091 1, 65050, David Slowinski, 1985,Mersenne 31. 30, 2 132049 -1, 39751, David Slowinski, 1983, Mersenne30. 29, 2
    http://jamesjoe.51.net/basic/basic02.html

    ftp://entropia.com/gimps/prime95.zip
    n -1ÊÇËØÊý,¶ø¶ÔÓÚÆäËûËùÓÐСÓÚ257µÄÊýn,MnÊǺÏÊý.µ«ÊÇ,ÕâÀï³öÏÖÁË5¸ö´íÎó,M67,M257²»ÊÇËØÊý,¶øM61,M89,M107ÊÇËØÊý.ÏÔÈ»,ҪʹMnÊÇËØÊý,n±¾Éí±ØÐëÊÇËØÊý,µ«ÊÇ·´¹ýÀ´,nÊÇËØÊý,MnÈ´²»Ò»¶¨ÊÇËØÊý,ÀýÈçËäÈ»11ÊÇËØÊý,¿ÉÊÇM11=2047=23X89ÊǺÏÊý.ÓÉ·ÑÂíС¶¨ÀíÖªµÀ,ÈôpΪËØÊý,¶øa²»ÊÇpµÄ±¶Êý,Ôòa p-1 -1ºãÊÇpµÄ±¶Êý.ÀýÈç2 -1ÊÇ7µÄ±¶Êý.µ«ÊÇÒ²ÓпÉÄÜ´æÔÚ±Èp-1¸üСµÄÊýb¾ÍÄÜʹµ2 b -1ÊÇpµÄ±¶ÊýÁË,ÀýÈç2 -1¾ÍÄܹ»±»7Õû³ýÁË.ºÜÈÝÒ×Ö¤÷p-1ÊÇbµÄ±¶Êý,¼´p=rb+1.±íæÉÏ¿´À´¶ÔÓÚÑ°ÕÒMpµÄÒò×Ó·ÑÂíС¶¨ÀíÒ»µã¶ùÒ²°ï²»ÉϦ,µ«ÊÇÒѾ­Ö¤÷,¶ÔÓÚµ×Êý2À´Ëµ,Ö¸Êýp-1¿É³ýÒÔ2¶ø²»Ó°Ïì¸ÊýÓpÈ¥³ýʱµÄ¿É³ýÐÔ,ÈçpÔÚ³ýÒÔ8ʱÓàÊýΪ1»òΪ7µÄ»°,»»ÑÔÖ®,¼´ÆäÐÎ״Ϊ8r+1»ò8r+7.´Ó¶ø¿ÉÒÔÍƳö: ¿ÉÒÔ±»pÕû³ý.ÔÚÇ°Ò»ÖÖÇé¿öÏÂ,˵÷2
    ÄÜÂú×ãÉÏÊöÌõ¼þµÄrÓënµÄÊýÖµ,¿ÉÒԲο´Ï±í: r sp -1ÊÇqµÄ±¶Êý,´Ó¶ø2 p -1Ò²ÊÇqµÄ±¶Êý,ÓÉÓÚq-1ÊÇżÊý,¶øpÊÇÆæÊý,ËùÒÔs±ØÊÇżÊý,Áîs=2k,ÔòÓÐ
    q=2kp+1,Õâ¾ÍÊÇ˵,ÐÎÈç2 p -1(pÊÇËØÊý)µÄÊý¾ßÓÐÐÎÈçq=2kp+1µÄÒò×Ó.ÀûÓÕâÒ»µã¿ÉÒÔÇó³ö·É­ÊýµÄÒò×Ó.ÀýÈçÒѾ­ÖªµÀ2 -1ÊǺÏÊý,ËüµÄÒò×Ó±ØÈ»ÊÇÐÎÈç2k*11+1=22k+1µÄÐÎʽ,k=1ʱ,µµ½23,¶ø23¹ûÈ»Äܹ»Õû³ý2 ÒѾ­Ñо¿³öһЩ·Ç³£ÇÉîµÄ°ì·¨À´´ó´óµØÏ÷¼õÊÔ̽µÄ¹¤×÷Á¿,1876Äê,·¨¹úÊýѧ¼ÒE.V.¬¿¨·¢÷ÁËÒ»ÖÖ²âÊÔ·¨¼ì²é·É­ÊýµÄËØÐÔ.1930Äê,À¹úÊýѧ¼ÒD.H.À³Ä¬¶Ô´Ë½øÐÐÁ˸Ľø,ʹ֮·Ç³£ÓÐЧ,ʵÓ.·½·¨ÈçÏÂ: ¶¨ÒåÊýÁÐ:u =4,u

    40. The Mad Cybrarian's Library: Free Online E-texts - Authors S-Sl
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