Challenging Perfect Number.docx

1、Challenging Perfect NumberChallenging Perfect Number Passage and pictures by Huang Hongtao who is the post-doctorate from Universit di Roma La Sapienza and does researches on computational mathematics On January 7 this year， a mathematician named Cooper from the University of Central Missouri Cooper

2、， through the project called Internet Mersenne Prime Search （GIMPS）， found the largest known perfect number 2 74，207，280 （2 74207281-1） which means that after 74，207，281th power of 2 minus 1 and then multiplied by 2 to the power of 74，207，280 （NOTE： is the computer language power symbol）. This numbe

3、r is the 49th perfect number human discovered during the past 2500 years and it has 44，677，235 digits； if it were printed in the ordinary size， the length will be 200 km！ Readers can imagine more about it . Even the well-known Australian mathematician Parker agrees that this is a tremendous scientif

4、ic achievement. Well， will it be an explorative story similar to the classic movie Good Will Hunting？ What kind of charm on earth does the perfect number boast， attracting so many mathematicians on after another to devote themselves？ What is Perfect Number？ Perfect number has some other names in Chi

5、nese whose definition is that the sum of all factors except itself equals the number itself. This definition seems to be confusing， so lets cite two examples， the two smallest perfect numbers are 6 whose all factors are 1，2，3，6 and 28 whose all factors are 1，2，4，7，14，28）， both of which are the sum o

6、f their all factors in addition to itself： 6 = 1 + 2 + 3 and 28 + 1 = 2 + 4 + 7 + 14. Due to some common magical properties， scientists entitle them with a wonderful name called the perfect number. For thousand years， the intriguing properties and incomparable charm of the perfect number has attract

7、ed many mathematicians and countless mathematic amateurs to explore it. In 17th century， French mathematician and philosopher Descartes publicly predicted， “The number of the perfect number will not be large， just like its not easy to find a perfect person. After a long period of time， people only f

8、ind 49 perfect numbers. This number of rare yet beautiful which is known as the diamonds in treasure house of number theory. Discovery of Mersenne Prime Later， the structure of 2P-1 using for finding the perfect number was named as Mersenne Prime in the mathematical field. It was named after Mersenn

9、e who is a French mathematician because his systematic and deep research into this special prime. Interestingly， the super primes found in the past century are almost Mersenne primes. Mersenne prime seems to be simple but actually its hard to explore. It requires not only profound theories and skill

10、ful techniques， but also complicated calculations and great amount of computation. In 1772 when Euler was blind， he still proved 231-1 （ie 2147483647） to be the eighth Mersenne prime by mental arithmetic. This prime with 10 digits was the largest known prime number at that time. The eighth perfect n

11、umber -230 （231-1） also emerged which was likely to be the biggest perfect number people found at that moment. Eulers perseverance and problem-solving skills were impressive. What French mathematician Laplace says maybe can represent our voice： Read Eulers writings. He is the teacher of everyone of

12、us. Against the backdrop where all the calculations and record must be written down by hands， no matter how hard people tried， they found only 12 Mersenne primes， that is to say， only 12 perfect numbers were discovered. However， the birth of computers greatly accelerated the process of research into

13、 Mersenne primes. For example， in 1952， the American mathematician Robinson compiled Lucas - Lehmer test into a computer program. By using the SWAC computer in a few months， they found five Mersenne primes： 2521-1，2607 -1，21279-1，22203-1 and 22281-1. Exploration into Mersenne prime is not only chall

14、enging. For the explorers， they can obtain a sense of pride. Perhaps this is why there are countless mathematicians willing to devote themselves into it. At 20：00 on June 2， 1963， when the first 23rd Mersenne prime 211213-1 was found by large computer， the American Broadcasting Corporation （ABC） int

15、errupted its regular broadcast and timely published this important message. All the students and faculties from the Department of Mathematics， University of Illinois staff who found the prime were very proud and at the same time， in order to let the world to share this great achievement， all the env

16、elopes sent from the department are covered with a 211213-1 is a prime number postmark， which was very crazy and fantastic. With the increase of the index P， the discovery of every Mersenne prime encounters tremendous hardships. However， the professional mathematicians and amateur mathematics lovers were not tired of this and even tried to take the lead. On February 23， 1979， when Vinski and Nelson who are the computer exp