PerfectNumbersandMersennePrimes-beingamathematician完全数梅森素数-beingamathematician

Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,W,2004 All rights reserved,5,7,2,1,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 will 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,12,18,15,Abundant,Abundant,Deficient,6,1,2,3,6,1+2+3=6,Check the factors of the numbers on your list to see if they are,Abundant,Deficient,or,Perfect,.Can you find P,2,the second perfect number?,6,=,6,Perfect Number,Perfect Number,P,1,=6,Factors,A/D/P,Factors,A/D/P,1,25,2,26,3,27,4,28,5,29,6,1,2,3,P,30,7,31,8,32,9,33,10,34,11,35,12,1,2,3,4,6,A,36,13,37,14,38,15,1,3,5,D,39,16,40,17,41,18,1,2,3,6,9,A,42,19,43,20,44,21,45,22,46,23,47,24,48,Also consider:,1.The distribution of abundant and deficient numbers.,2.Numbers with the fewest factors.,3.Numbers with the most factors.,Factors,A/D/P,Factors,A/D/P,1,25,1,5,D,2,26,1,2,13,D,3,27,1,3,9,D,4,28,1,2,4,7,14,P,5,29,1,Prime,6,1,2,3,P,30,1,2,3,5,6,10,15,A,7,1,Prime,31,1,Prime,8,1,2,4,D,32,1,2,4,8,16,D,9,1,3,D,33,1,3,11,D,10,1,2,5,D,34,1,2,17,D,11,1,Prime,35,1,5,7,D,12,1,2,3,4,6,A,36,1,2,3,4,6,9,12,18,A,13,1,Prime,37,1,Prime,14,1,2,7,D,38,1,2,19,D,15,1,3,5,D,39,1,3,13,D,16,1,2,4,8,D,40,1,2,4,5,8,10,20,A,17,1,Prime,41,1,Prime,18,1,2,3,6,9,A,42,1,2,3,6,7,14,21,A,19,1,Prime,43,1,Prime,20,1,2,4,5,10,A,44,1,2,4,11,22,D,21,1,3,7,D,45,1,3,5,9,15,D,22,1,2,11,D,46,1,2,23,D,23,1,Prime,47,1,Prime,24,1,2,3,4,6,8,12,A,48,1,2,3,4,6,8,12,16,24,A,Obviously the prime numbers have 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.,Factors,A/D/P,Factors,A/D/P,1,25,1,5,D,2,26,1,2,13,D,3,27,1,3,9,D,4,28,1,2,4,7,14,P,5,29,1,Prime,6,1,2,3,P,30,1,2,3,5,6,10,15,A,7,1,Prime,31,1,Prime,8,1,2,4,D,32,1,2,4,8,16,D,9,1,3,D,33,1,3,11,D,10,1,2,5,D,34,1,2,17,D,11,1,Prime,35,1,5,7,D,12,1,2,3,4,6,A,36,1,2,3,4,6,9,12,18,A,13,1,Prime,37,1,Prime,14,1,2,7,D,38,1,2,19,D,15,1,3,5,D,39,1,3,13,D,16,1,2,4,8,D,40,1,2,4,5,8,10,20,A,17,1,Prime,41,1,Prime,18,1,2,3,6,9,A,42,1,2,3,6,7,14,21,A,19,1,Prime,43,1,Prime,20,1,2,4,5,10,A,44,1,2,4,11,22,D,21,1,3,7,D,45,1,3,5,9,15,D,22,1,2,11,D,46,1,2,23,D,23,1,Prime,47,1,Prime,24,1,2,3,4,6,8,12,A,48,1,2,3,4,6,8,12,16,24,A,For Homework:,There is only one,weird number,below 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 knowledge:Aristotle.,THE SCHOOL of ATHENS,(Raphael)1510-11,Pythagoras,Euclid,Plato,Aristotle,Socrates,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.),P,1,=6,P,2,=28,P,3,=496,P,4,=8128,1+2+3=,6,1+2+4+7+14=,28,1+2+4+8+16+31+62+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 P,5,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.),P,1,=6,P,2,=28,P,3,=496,P,4,=8128,How many digits would you reasonably expect P,5,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 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.),P,1,=6,P,2,=28,P,3,=496,P,4,=8128,The Greeks never managed to find it.This elusive number turned up in medieval Europe in an anonymous manuscript and it was an,8 digit,number,P,5,=33 550 336,P,5,=33 550 336,(1456 Not Known),8 digits,P,6,=8 589 869 056,(1588 Cataldi),10 digits,P,7,=137 438 691 328,(1588 Cataldi),12 digits,P,8,=2 305 843 008 139 952 128,(1772 Euler),19 digits,P,9,=2 658 455 991 569 831 744 654 692 615 953 842 176,(1883 Pervushin),37 digits,P,10,=191 561 942 608 236 107 294 793 378 084 303 638 130 997 321 548 169 216,(1911:Powers),54 digits,P,11,=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,P,12,=14 474 011 154 664 524 427 946 373 126 085 988 481 573 677 491,474 835 889 066 354 349 131 199 1