PerfectNumbersandMersennePrimes-beingamathematician完全数梅森素数-beingamathematician.ppt

1、Perfect Numbers,Abundant Numbers,Deficient Numbers,Weird Numbers,Proper Divisor: The proper divisors of a number are all its divisors (factors) excluding the number itself,Taking 36 as an example: Its proper divisors are 1, 2, 3, 4, 6, 9, 1 2, and 18 but not 36,In the investigation that follows we w

2、ill only consider proper divisors,Mersenne Primes,12,Factors,1, 2, 3, 4, 6, 12,1 + 2 + 3 + 4 + 6 = 16,16 12,18,1, 2, 3, 6, 9, 18,1 + 2 + 3 + 6 + 9 = 21,21 18,15,1, 3, 5, 15,1 + 3 + 5 = 9,9 15,Abundant Number,Abundant Number,Deficient Number,6,1, 2, 3, 6,1 + 2 + 3 = 6,Check the factors of the numbers

3、 on your list to see if they are Abundant, Deficient or Perfect. Can you find P2, the second perfect number,6 = 6 Perfect Number,Also consider: 1. The distribution of abundant and deficient numbers. 2. Numbers with the fewest factors. 3. Numbers with the most factors,Obviously the prime numbers have

4、 only one factor,There are more deficient numbers than abundant numbers and all the abundant numbers are even,Once you have done the “Product of Primes” presentation you may be able to see why numbers such as 24, 36, 40, and 48 have lots of factors,For Homework: There is only one weird number below

5、100 can you find it? A weird number is an abundant number that cannot be written as the sum of any subset of its divisors,As an example: 20 is not weird since it can be written as 1 + 4 + 5 + 10 and 36 is not weird since it can be written as 18 + 12 + 6 or 9 + 6 + 18 + 3,All Men by nature desire kno

6、wledge”: Aristotle,THE SCHOOL of ATHENS (Raphael) 1510 -11,Pythagoras,Euclid,The Mathematicians of Ancient Greece,Pythagoras of Samos (570 500 BC.,Euclid of Alexandria(325 265 BC.,Archimedes of Syracuse (287 212 BC.,Eratosthenes of Cyene (275-192 BC.,P3 = 496,P4 = 8128,1 + 2 + 4 + 8 + 16 + 31 + 62 +

7、 124 + 248 = 496,1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 = 8128,The mathematicians of Ancient Greece knew the first 4 perfect numbers and the search was on for the P5,The Mathematicians of Ancient Greece,Pythagoras of Samos (570 500 BC.,Euclid of Alexandria(325 265 BC.,Ar

8、chimedes of Syracuse (287 212 BC.,Eratosthenes of Cyene (275-192 BC.,P3 = 496,P4 = 8128,How many digits would you reasonably expect P5 to have and what is the largest number that you can make with this many digits,Five digits seems reasonable considering the digit sequence 1,2,3,4 and 99,999 is the

9、highest 5 digit number,The Mathematicians of Ancient Greece,Pythagoras of Samos (570 500 BC.,Euclid of Alexandria(325 265 BC.,Archimedes of Syracuse (287 212 BC.,Eratosthenes of Cyene (275-192 BC.,P3 = 496,P4 = 8128,The Greeks never managed to find it. This elusive number turned up in medieval Europ

10、e in an anonymous manuscript and it was an 8 digit number,P5 = 33 550 336,P5 = 33 550 336 (1456 Not Known) 8 digits P6 = 8 589 869 056 (1588 Cataldi) 10 digits P7 = 137 438 691 328 (1588 Cataldi) 12 digits P8 = 2 305 843 008 139 952 128 (1772 Euler) 19 digits P9 = 2 658 455 991 569 831 744 654 692 6

11、15 953 842 176 (1883 Pervushin) 37 digits P10 = 191 561 942 608 236 107 294 793 378 084 303 638 130 997 321 548 169 216 (1911: Powers) 54 digits P11 =13 164 036 458 569 648 337 239 753 460 458 722 910 223 472 318 386 943 117 783 728 128 (1914 Powers) 65 digits P12 =14 474 011 154 664 524 427 946 373

12、 126 085 988 481 573 677 491 474 835 889 066 354 349 131 199 152 128 (1876 Edouard Lucas) 77 digits P13=2356272345726734706578954899670990498847754785839260071014302052892578043215543382498428777152427010394496918664028644534175975063372831786222397303655396026005613602555664625032701752803383143979

13、0236838624033171435922356643219703101720713163527487298747400647801939587165936401087419375649057918549492160555646 976 (1952 Robinson) 314 digits,P1= 6 P2= 28 P3= 496 P4 = 8128 P5 = ,a 5 digit number,As you can see perfect numbers are extremely rare. There are only 41 known perfect numbers. The lar

14、gest P41 (discovered in 2004) has 14 471 465 digits,There is a strong link between perfect numbers and other numbers called Mersenne Primes. Mersenne primes are as rare as perfect numbers, M41 is the largest known prime just as P41 is the largest known perfect number. In fact every time someone find

15、s a new one they can calculate the resulting perfect number by use of a formula. For every Mersenne Prime there is a corresponding Perfect number. The 41st Mersenne Prime has 7 235 733 digits. Each perfect number has roughly double the number of digits as its corresponding Mersenne prime. Each time

16、one is discovered it automatically becomes the worlds largest known prime,There is a $100,000 prize for the first person that discovers a prime with over 10 million digits and it could be you,Largest Prime Discovered,Monday 7th June 2004,A scientist has used his computer to find the largest prime nu

17、mber found so far.- written out, it would stretch for 25 km. The number is the Mersenne prime 224 036 583 1. The new figure identified by Josh Findley contains 7,235,733 digits and would take someone the best part of 6 weeks to write out by hand. Mr Findley was taking part in a mass computer project

18、 known as GIMPS (Great Internet Mersenne Prime Search). Gimps is closing in on the $100 000 prize for the first person to find a 10-milliondigit-prime! “An award winning prime could be mere weeks away or as much as a few years away” said GIMPS founder George Woltman,41st Mersenne Prime Found,Primes

19、are the building blocks of all whole numbers,Historically searching for Mersenne primes has been used to test computer hardware. The free GIMPS programme used by Findley has identified hidden hardware problems in many computers. He used a 2.4 GHz Pentium 4 Windows XP computer running for 14 days to

20、prove the number was prime. Prime numbers are mainly of interest to mathematicians that study the branch of mathematics called Number Theory but they are becoming important in cryptography and may eventually lead to uncrackable codes,Mersenne Primes,A Mersenne number is any number of the form 2n 1.

21、The first 12 of these are shown below,21 1 = 1,22 1 = 3,23 1 = 7,24 1 = 15,25 1 = 31,26 1 = 63,27 1 = 127,28 1 = 255,29 1 = 511,210 1 = 1023,211 1 = 2047,212 1 = 4095,Early mathematicians thought that if n was prime then 2n -1 was also prime,Mersenne Primes,A Mersenne number is any number of the for

22、m 2n 1. The first 12 of these are shown below,21 1 = 1,22 1 = 3,23 1 = 7,24 1 = 15,25 1 = 31,26 1 = 63,27 1 = 127,28 1 = 255,29 1 = 511,210 1 = 1023,211 1 = 2047,212 1 = 4095,Early mathematicians thought that if n was prime then 2n -1 was also prime,In 1536 Hudalricus Regius showed that 211 -1 = 23

23、x 89 and so was not prime,Mersenne Primes,A Mersenne Prime is any prime number of the form 2n 1. were n is prime,A French monk called Marin Mersenne stated in one of his books in 1644 that for the primes: n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257 that 2n -1 would be prime and that all other p

24、ositive integers less than 257 would yield only composite (non-prime) numbers. He stated this even though he could not possibly have checked such huge numbers. He was making a conjecture,22 1 = 3,23 1 = 7,25 1 = 31,27 1 = 127,1588 - 1644,After this all such numbers that were prime became known as Me

25、rsenne Primes. In subsequent years various mathematicians showed that his conjecture was not correct,Mersenne Primes,22 1 = 3,23 1 = 7,25 1 = 31,27 1 = 127,1588 - 1644,213 1,217 1,219 1,231 1,2127 1,n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127,A Mersenne Prime is any number of the form 2n 1. wer

26、e n is prime and produces a prime number,Completed List,261 1,289 1,2107 1,It was 1947 before all the numbers on Mersennes original list had been checked,Mersenne Primes and Perfect Numbers,22 1 = 3,23 1 = 7,25 1 = 31,27 1 = 127,1588 - 1644,213 1,217 1,219 1,231 1,2127 1,A Mersenne Prime is any numb

27、er of the form 2n 1. were n is prime and produces a prime number,261 1,289 1,2107 1,There is a formula linking a Mersenne Prime to its corresponding perfect number by multiplication. Use the table below to help you find it,Mersenne Primes and Perfect Numbers,22 1 = 3,23 1 = 7,25 1 = 31,27 1 = 127,15

28、88 - 1644,213 1,217 1,219 1,231 1,2127 1,A Mersenne Prime is any number of the form 2n 1. were n is prime and produces a prime number,261 1,289 1,2107 1,There is a formula linking a Mersenne Prime to its corresponding perfect number by multiplication. Use the table below to help you find it,Mersenne

29、 Primes and Perfect Numbers,22 1 = 3,23 1 = 7,25 1 = 31,27 1 = 127,1588 - 1644,213 1,217 1,219 1,231 1,2127 1,A Mersenne Prime is any number of the form 2n 1. were n is prime and produces a prime number,261 1,289 1,2107 1,There is a formula linking a Mersenne Prime to its corresponding perfect numbe

30、r by multiplication. Use the table below to help you find it,Mersenne Primes and Perfect Numbers,22 1 = 3,23 1 = 7,25 1 = 31,27 1 = 127,1588 - 1644,213 1,217 1,219 1,231 1,2127 1,A Mersenne Prime is any number of the form 2n 1. were n is prime and produces a prime number,261 1,289 1,2107 1,If 2n -1

31、is a Mersenne prime then 2n 1 x 2n-1 is a perfect number. Check this for the first few,Mersenne Primes and Perfect Numbers,22 1 = 3,23 1 = 7,25 1 = 31,27 1 = 127,1588 - 1644,213 1,217 1,219 1,231 1,2127 1,A Mersenne Prime is any number of the form 2n 1. were n is prime and produces a prime number,261 1,289 1,21