一群學生找到第40個Mersenne質數，請參考！
GIMPS has discovered six Mersenne primes so far. On November 17, 2003 Michael Shafer discovered the 40th known Mersenne prime, 2^20996011-1. This is the largest known prime number. On November 14th 2001, Michael Cameron found the previous world record prime, 2^13466917-1. On June 1st 1999, Nayan Hajratwala found the prime, 2^6972593-1. On January 27th 1998, Roland Clarkson found the prime, 2^3021377-1. On August 24th 1997, Gordon Spence discovered the prime 2^2976221-1. In November 1996, Joel Armengaud found GIMPS' first prime, 2^1398269-1. Could you be next?
As a collector I get excited when we can announce a new prime. After over 2000 years of looking for Mersenne primes, a 40th has been found: 2^20996011-1. Of course it was GIMPS (the Great Internet Mersenne Prime Search) again with its over 210,000 networked personal computers run by some 60,000 volunteers. They have found all of the recent Mersennes.
This prime was actually found by GIMPS volunteer Michael Shafer, a 26 year-old chemical engineering graduate student at Michigan State University, using a simple Dell computer in a university lab. The excellent press announcement on the GIMPS site carefully lays out the details, so I will omit them here. Instead I will address the reporter's usual problem--what is the angle? Is there a story here?
But first let me say again "wow! what a prime!" Congratulations to GIMPS (especially its founder George Woltman), Michael, and all of those involved. Thank-you again for giving an old prime collector something else to treasure. I wish it was me!
Will reporters find a story here?
That is a hard question. Reporters often like sex, or destruction, or violence. Sorry, none of that here. They hope for something radically new or earth shaking. Again, none of that here. This prime is roughly where expected, and adding one more example does not alter mathematics.
Sure, the number is big, but this behemoth has 6,320,430 decimal digits and is so large it has no physical counterpart. Most people think the US National Debt is big, and that has a mere 12 digits. The number of atoms in the universe has less than 90 digits. A reporter can't build a story on size alone anymore. Folks just don't know what big is.
http://www.utm.edu/research/primes/largest.html
http://www.mersenne.org/history.htm
孫文先敬上
GIMPS has discovered six Mersenne primes so far. On November 17, 2003 Michael Shafer discovered the 40th known Mersenne prime, 2^20996011-1. This is the largest known prime number. On November 14th 2001, Michael Cameron found the previous world record prime, 2^13466917-1. On June 1st 1999, Nayan Hajratwala found the prime, 2^6972593-1. On January 27th 1998, Roland Clarkson found the prime, 2^3021377-1. On August 24th 1997, Gordon Spence discovered the prime 2^2976221-1. In November 1996, Joel Armengaud found GIMPS' first prime, 2^1398269-1. Could you be next?
As a collector I get excited when we can announce a new prime. After over 2000 years of looking for Mersenne primes, a 40th has been found: 2^20996011-1. Of course it was GIMPS (the Great Internet Mersenne Prime Search) again with its over 210,000 networked personal computers run by some 60,000 volunteers. They have found all of the recent Mersennes.
This prime was actually found by GIMPS volunteer Michael Shafer, a 26 year-old chemical engineering graduate student at Michigan State University, using a simple Dell computer in a university lab. The excellent press announcement on the GIMPS site carefully lays out the details, so I will omit them here. Instead I will address the reporter's usual problem--what is the angle? Is there a story here?
But first let me say again "wow! what a prime!" Congratulations to GIMPS (especially its founder George Woltman), Michael, and all of those involved. Thank-you again for giving an old prime collector something else to treasure. I wish it was me!
Will reporters find a story here?
That is a hard question. Reporters often like sex, or destruction, or violence. Sorry, none of that here. They hope for something radically new or earth shaking. Again, none of that here. This prime is roughly where expected, and adding one more example does not alter mathematics.
Sure, the number is big, but this behemoth has 6,320,430 decimal digits and is so large it has no physical counterpart. Most people think the US National Debt is big, and that has a mere 12 digits. The number of atoms in the universe has less than 90 digits. A reporter can't build a story on size alone anymore. Folks just don't know what big is.
http://www.utm.edu/research/primes/largest.html
http://www.mersenne.org/history.htm
孫文先敬上
回應
Fernando
Re: 目前找到的最大質數
2003-12-03 19:48不都是用2^n-1的方式去找嗎.但要驗算就難了.
所以小弟我太佩服他們了.
我記的國中時看到書上寫的最大質數是2^3021377-1,沒想到這麼快就被破啦.
訪客 (未驗證)
Re: 目前找到的最大質數
2003-12-03 21:28人們利用將兩個很大的質數相乘作為公開鑰，運用在密碼學上。以下摘錄九章出版《密碼學》上的一段相關文章。
孫文先敬上
RSA與公開鑰密碼系統
在1977年，麻省理工學院研究員Ronald Rivest、Adi Shamir及Leonard Adleman寫下了現代密碼學上最重要的一篇文章。文中提到的RSA密碼算法，該算法等同於公開鑰密碼，是使用一個非對稱的密碼，當史丹福大學的W. Diffie在1975年發表該概念時，DH交換密鑰協議還沒有面世。非對稱密碼的概念，主要是容陰策枯者參與加密的過程。
(1) 無需公共密鑰的密碼
假設A和B生活在一個十分落後的地方，郵政服務極為腐敗。任何容易打開的包裹內的物品都會被偷走，但上了鎖的箱子卻可以安全無恙到達目的地。現在A想要把一只貴重的手錶郵寄給B，如果隨便投寄的話，必然會於途中被盜。如果A把手錶放在一個上了鎖的箱內投寄，則必可平安抵達；但B郤沒有開鎖的鑰匙，結果亦不能取出手錶。
從這個例子可見，公共密鑰並不是保障訊息安全傳送的必要條件。無需公共密鑰的非對稱密碼，為解決分發密鑰的問題帶來了新希望。麻省理工的科學家找到落實該概念的方法，其中使用公開鑰的加密運算十分容易，若不知道私有鑰卻無法解密。RSA所用的暗門單向函數，是利用因子分解比乘法困難得多的事實。設p = 47,951及q = 32,257。利用計算機不消幾秒鐘便可把n = p ´ q = 1,546,755,407算出來。反過來說，若要將n = 1,546,755,407因子分解成p和q，所需的時間何止百倍。
當1977年8月號的科學美國人(Scientific American)雜誌首次刊出關於RSA算法的文章時，作者Tony Gardner挑戰讀者因子分解一個129位的n值，稱為RSA-129。該n值是加密一篇密文的公開鑰，要把密文破譯便需要因子分解n，因為n的質數因子就是解密所需的私有鑰。文中懸賞100美元給予首個把n因子分解的人。直至1994年，由六百個人仕組合而成的隊伍終於奪得了賞金。他們在1993年開始整合了超過1600台電腦，以八個月的光陰找出二個分別為64位及65位的因數。
以今天的安全標準來說RSA-129是一個很小的數。現在RSA算法使用的n值約有200個位，而在一些重要的銀行交易裏選用的n值通常約有300個位；因此，就算結合全世界的所有電腦，於天荒地老之時亦不可以把RSA密碼破譯呢！要攻破RSA有兩種可能的方法：第一是算法上的突破，第二是計算能力的改進。未來是否有數學家想出一個突破性的因子分解算法仍是未知之數；在過去二千多年，無數的數學家曾在這方面下過奶牷A但還未找到一個較有效的算法。
電腦的計算能力每十八個月便增強一倍，到了2062年，因子分解一個300位的十進數，仍需要約2.4 ´ 1014 ¸ 240年，即大約是220年。只要增加密鑰的長度，即n的十進制位數的數目，便可加強保安的能力。
既然說RSA這麼強，密碼學的研究是否基本上終結了呢？其實不然，現時有不少研究員把精力投放在一些突破性的概念，如量子物理學上。事實上，量子物理學可說是資訊科技的基石，若量子電腦面世，其計算能力將會比現時的電腦成幾十甚至幾千倍幾何級數地增加；由它來破解像RSA這類建基於暗門單向函數的密碼，便會易如反掌。另一方面，量子物理學亦提出了一個可能是真正不可破解的資料傳送方案。
Fernando
Re: 目前找到的最大質數
2003-12-03 21:56其實數學真的奇妙ㄟ.有些乍看之下已經是質數的數,卻還可以分解.如果是超過幾10位的數,那還真必須用到電腦了.眞希望我以後能破解質數之謎阿.
訪客 (未驗證)
Re: 目前找到的最大質數
2004-02-18 00:52資訊科技的進展有可能會帶動算法的突破, 2062年的估計可能需要修正...