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Whiteboardmaths.com 7 2 1 5 © 2004 All rights reserved 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. Perfect Numbers Abundant Numbers Deficient Numbers Mersenne Primes Weird Numbers Factors: 12 1, 2, 3, 4, 6, 12 18 1, 2, 3, 6, 9, 18 15 1 + 2 + 3 + 4 + 6 = 16 16 > 12 Abundant Number 1 + 2 + 3 + 6 + 9 = 21 Abundant Number 21 > 18 1, 3, 5, 15 1+3+5=9 9 < 15 Deficient Number 12 18 15 Abundant Abundant Deficient 6 1, 2, 3, 6 P1 = 6 1+2+3=6 6 = 6 Perfect Number Perfect Number Check the factors of the numbers on your list to see if they are Abundant, Deficient or Perfect. Can you find P2, the second perfect number? Factors A/D/P Factors 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 A/D/P Also consider: 1. The distribution of abundant and deficient numbers. 2. Numbers with the fewest factors. 3. Numbers with the most factors. There are more Factors deficient numbersA/D/P 1 than abundant numbers and all the 2 abundant numbers are even. 3 1, 5 D 26 1, 2, 13 D 27 1, 3, 9 D 28 1, 2, 4, 7, 14 P 29 1 30 1, 2, 3, 5, 6, 10, 15 31 1 32 1, 2, 4, 8, 16 D 33 1, 3, 11 D 34 1, 2, 17 D 35 1, 5, 7 D A 36 1, 2, 3, 4, 6, 9, 12, 18 A Prime 37 1 4 1 A/D/P 25 Obviously the prime numbers have 5 only one factor. 6 1, 2, 3 P 7 Factors Prime Once you have done the “Product of 8 1, 2, 4 D Primes” 9 1, 3 presentation you may be D able to why numbers such as 24, 10 1, 2,see 5 D 11 40, 1 and 48 have lots of factors. Prime 36, Prime A Prime 12 1, 2, 3, 4, 6 13 1 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 18 1, 2, 3, 6, 9 A 42 1, 2, 3, 6, 7, 14, 21 19 1 Prime 43 1 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 24 1, 2, 3, 4, 6, 8, 12 A 48 1, 2, 3, 4, 6, 8, 12, 16, 24 Prime Prime A Prime Prime A For Homework:Factors A/D/P 1 1, 2, 4 A/D/P 25 1, 5 D 26 1, 2, 13 D 27 1, 3, 9 D 28 1, 2, 4, 7, 14 P 29 1 30 1, 2, 3, 5, 6, 10, 15 31 1 32 1, 2, 4, 8, 16 D 33 1, 3, 11 D 34 1, 2, 17 D 35 1, 5, 7 D 36 1, 2, 3, 4, 6, 9, 12, 18 A Prime 37 1 2 There is only one weird number 3 below 100 can you find it? A weird 4 number is an abundant number that 5 cannot be written as the sum of any 6 1, 2, 3 P subset of its divisors. 7 1 Prime 8 Factors D As 20 is not weird since 9 an1,example: 3 D it can written as 1 + 4 + 5 + 10 andD 10 be 1, 2, 5 11 is not 1 Prime 36 weird since it can be written 3, 4,+6 6 or 9 + 6 + 18 + 3 A as12 18 1,+2,12 Prime A Prime 13 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 18 1, 2, 3, 6, 9 A 42 1, 2, 3, 6, 7, 14, 21 19 1 Prime 43 1 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 24 1, 2, 3, 4, 6, 8, 12 A 48 1, 2, 3, 4, 6, 8, 12, 16, 24 Prime A Prime Prime A THE SCHOOL of ATHENS (Raphael) 1510 -11 Socrates Pythagoras Plato Aristotle Euclid “All Men by nature desire knowledge”: Aristotle. The Mathematicians of Ancient Greece. Pythagoras of Samos (570 – 500 BC.) P1 = 6 P2 = 28 P3 = 496 P4 = 8128 Euclid of Alexandria (325 – 265 BC.) Archimedes of Syracuse (287 – 212 BC.) Eratosthenes of Cyene (275-192 BC.) 1+2 + 3 mathematicians =6 The of Ancient Greece knew the first 4 perfect numbers and 1 + 2 + 4 + 7 + 14 = 28 the search was on for the P5 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. Pythagoras of Samos (570 – 500 BC.) P1 = 6 P2 = 28 P3 = 496 P4 = 8128 Euclid of Alexandria (325 – 265 BC.) Archimedes of Syracuse (287 – 212 BC.) Eratosthenes of Cyene (275-192 BC.) 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 highest 5 digit number. The Mathematicians of Ancient Greece. Pythagoras of Samos (570 – 500 BC.) P1 = 6 P2 = 28 P3 = 496 P4 = 8128 Euclid of Alexandria (325 – 265 BC.) Archimedes of Syracuse (287 – 212 BC.) Eratosthenes of Cyene (275-192 BC.) 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 P5 = 33 550 336 P1= 6 P2= 28 P3= 496 P4 = 8128 P5 = ? (a 5 digit number?) 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 = 2 658 455 991 569 831 744 654 692 615 953 842 176 (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 =14 474 011 154 664 524 427 946 373 126 085 988 481 573 677 491 474 835 889 066 354 349 131 199 152 128 (1876 Edouard Lucas) 77 digits P13=23562723457267347065789548996709904988477547858392600710 143020528925780432155433824984287771524270103944969186640286 44534175975063372831786222397303655396026005613602555664625 032701752803383143979023683862403317143592235664321970310172 071316352748729874740064780193958716593640108741937564905791 8549492160555646 976 (1952 Robinson) 314 digits As you can see perfect numbers are extremely rare. There are only 41 known perfect numbers. The largest 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 finds 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 one is discovered it automatically becomes the world’s 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! Monday 7th June 2004 Largest Prime Discovered! A scientist has used his computer to find the largest prime number 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 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-million–digitprime! “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 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 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. The first 12 of these are shown below. 21 – 1 = 1 22 – 1 = 3 26 – 1 = 63 210 – 1 = 1023 23 – 1 = 7 27 – 1 = 127 24 – 1 = 15 28 – 1 = 255 211 – 1 = 2047 25 – 1 = 31 29 – 1 = 511 212 – 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 form 2n – 1. The first 12 of these are shown below. 21 – 1 = 1 22 – 1 = 3 26 – 1 = 63 210 – 1 = 1023 23 – 1 = 7 27 – 1 = 127 24 – 1 = 15 28 – 1 = 255 211 – 1 = 2047 25 – 1 = 31 29 – 1 = 511 212 – 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 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. 22 – 1 = 3 23 – 1 = 27 – 1 = 127 25 – 1 = 7 31 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 1588 - 1644 that 2n -1 would be prime and that all other positive 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. After this all such numbers that were prime became known as Mersenne Primes. In subsequent years various mathematicians showed that his conjecture was not correct. Mersenne Primes A Mersenne Prime is any number of the form 2n – 1. were n is prime and produces a prime number.. 22 – 1 = 3 23 – 1 = 7 219 25 – 1 = 31 – 1 231 – 1 27 – 1 = 127 261 – 1 289 – 1 213 – 1 2107 – 1 217 – 1 2127 – 1 Mersenne’s List n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257 1588 - 1644 Two of the numbers on Mersenne’s list did not generate Mersenne primes and there were others that he had missed off. Completed List n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, It was 1947 before all the numbers on Mersenne’s original list had been checked Mersenne Primes and Perfect Numbers A Mersenne Prime is any number of the form 2n – 1. were n is prime and produces a prime number.. 22 – 1 = 3 23 – 1 = 7 219 25 – 1 = 31 – 1 231 – 1 213 – 1 27 – 1 = 127 261 – 1 289 – 1 2107 – 1 217 – 1 2127 – 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. 1588 - 1644 Perfect Number n 2n -1 2 3 x ? 6 3 7 x ? 28 5 31 x ? 496 7 127 x ? 8128 Mersenne Primes and Perfect Numbers A Mersenne Prime is any number of the form 2n – 1. were n is prime and produces a prime number.. 22 – 1 = 3 23 – 1 = 7 219 25 – 1 = 31 – 1 231 – 1 213 – 1 27 – 1 = 127 261 – 1 289 – 1 2107 – 1 217 – 1 2127 – 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. 1588 - 1644 Perfect Write as a Number power of 2 n 2n -1 2 3 x 2 6 3 7 x 4 28 5 31 x 16 496 7 127 x 64 8128 Mersenne Primes and Perfect Numbers A Mersenne Prime is any number of the form 2n – 1. were n is prime and produces a prime number.. 22 – 1 = 3 23 – 1 = 7 219 25 – 1 = 31 – 1 231 – 1 213 – 1 27 – 1 = 127 261 – 1 289 – 1 2107 – 1 217 – 1 2127 – 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. 1588 - 1644 Perfect Write as a Number power of 2 n 2n -1 2 3 x 2 6 21 3 7 x 4 28 22 5 31 x 16 496 24 7 127 x 64 8128 26 Mersenne Primes and Perfect Numbers A Mersenne Prime is any number of the form 2n – 1. were n is prime and produces a prime number.. 22 – 1 = 3 23 – 1 = 7 219 25 – 1 = 31 – 1 231 – 1 213 – 1 27 – 1 = 127 261 – 1 289 – 1 2107 – 1 217 – 1 2127 – 1 If 2n -1 is a Mersenne prime then 2n – 1 x 2n-1 is a perfect number. Check this for the first few. 1588 - 1644 Perfect Write as a Number power of 2 n 2n -1 2 3 x 2 6 21 3 7 x 4 28 22 5 31 x 16 496 24 7 127 x 64 8128 26 Mersenne Primes and Perfect Numbers A Mersenne Prime is any number of the form 2n – 1. were n is prime and produces a prime number.. 22 – 1 = 3 23 – 1 = 7 219 25 – 1 = 31 – 1 231 – 1 27 – 1 = 127 261 – 1 213 – 1 289 – 1 2107 – 1 217 – 1 2127 – 1 If 2n -1 is a Mersenne prime then 2n – 1 x 2n-1 is a perfect number. Check this for the first few. 2 2 – 1 x 21 = 3 x 2 = 6 23 – 1 x 22 = 7 x 4 = 28 1588 - 1644 25 – 1 x 24 = 31 x 16 = 496 27 – 1 x 26 = 127 x 64 = 8128 Mersenne Primes and Perfect Numbers A Mersenne Prime is any number of the form 2n – 1. were n is prime and produces a prime number.. 22 – 1 = 3 23 – 1 = 7 219 25 – 1 = 31 – 1 231 – 1 27 – 1 = 127 261 – 1 289 – 1 213 – 1 2107 – 1 217 – 1 2127 – 1 NEWS FLASH 4th September 2006 44th Mersenne Prime Found. 232 582 657 – 1 has 9,808,358 digits 1588 - 1644 The $100 000 prize for the world’s first 10 million digit prime is still on but you need to be quick. Mersenne Primes and Perfect Numbers A Mersenne Prime is any number of the form 2n – 1. were n is prime and produces a prime number.. 22 – 1 = 3 23 – 1 = 7 219 25 – 1 = 31 – 1 231 – 1 27 – 1 = 127 261 – 1 289 – 1 213 – 1 2107 – 1 217 – 1 2127 – 1 Research other information about Mersenne Primes and Perfect Numbers and don’t forget to join GIMPS. 1588 - 1644 http://www.mersenne.org/ Great Internet Mersenne Prime Search Factors A/D/P Factors 1 25 2 26 3 4 Worksheet 1 5 29 6 30 7 31 8 32 9 33 10 34 11 35 12 36 13 37 14 38 15 39 16 40 17 41 18 42 19 43 20 44 21 45 22 46 23 47 24 48 27 28 A/D/P Perfect Number n 2n -1 2 3 x 6 3 7 x 28 5 31 x 496 7 127 x 8128 n 2n -1 2 3 x 6 3 7 x 28 5 31 x 496 7 127 x 8128 Perfect Number Worksheet 2