2. Whiteboardmaths.com © 2004 All rights reserved 5 7 2 1

3. 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

4. 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 Deficient Number

5. 121815 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

6. FactorsA/D/PFactorsA/D/P 125 226 327 428 529 61, 2, 3P30 731 832 933 1034 1135 121, 2, 3, 4, 6A36 1337 1438 151, 3, 5D39 1640 1741 181, 2, 3, 6, 9A42 1943 2044 2145 2246 2347 2448 Also consider: 1. The distribution of abundant and deficient numbers. 2. Numbers with the fewest factors. 3. Numbers with the most factors.

7. FactorsA/D/PFactorsA/D/P 1251, 5D 2261, 2, 13D 3271, 3, 9D 4281, 2, 4, 7, 14P 5291Prime 61, 2, 3P301, 2, 3, 5, 6, 10, 15A 71Prime311Prime 81, 2, 4D321, 2, 4, 8, 16D 91, 3D331, 3, 11D 101, 2, 5D341, 2, 17D 111Prime351, 5, 7D 121, 2, 3, 4, 6A361, 2, 3, 4, 6, 9, 12, 18A 131Prime371Prime 141, 2, 7D381, 2, 19D 151, 3, 5D391, 3, 13D 161, 2, 4, 8D401, 2, 4, 5, 8, 10, 20A 171Prime411Prime 181, 2, 3, 6, 9A421, 2, 3, 6, 7, 14, 21A 191Prime431Prime 201, 2, 4, 5, 10A441, 2, 4, 11, 22D 211, 3, 7D451, 3, 5, 9, 15D 221, 2, 11D461, 2, 23D 231Prime471Prime 241, 2, 3, 4, 6, 8, 12A481, 2, 3, 4, 6, 8, 12, 16, 24A 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.

8. FactorsA/D/PFactorsA/D/P 1251, 5D 2261, 2, 13D 3271, 3, 9D 4281, 2, 4, 7, 14P 5291Prime 61, 2, 3P301, 2, 3, 5, 6, 10, 15A 71Prime311Prime 81, 2, 4D321, 2, 4, 8, 16D 91, 3D331, 3, 11D 101, 2, 5D341, 2, 17D 111Prime351, 5, 7D 121, 2, 3, 4, 6A361, 2, 3, 4, 6, 9, 12, 18A 131Prime371Prime 141, 2, 7D381, 2, 19D 151, 3, 5D391, 3, 13D 161, 2, 4, 8D401, 2, 4, 5, 8, 10, 20A 171Prime411Prime 181, 2, 3, 6, 9A421, 2, 3, 6, 7, 14, 21A 191Prime431Prime 201, 2, 4, 5, 10A441, 2, 4, 11, 22D 211, 3, 7D451, 3, 5, 9, 15D 221, 2, 11D461, 2, 23D 231Prime471Prime 241, 2, 3, 4, 6, 8, 12A481, 2, 3, 4, 6, 8, 12, 16, 24A 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

9. “All Men by nature desire knowledge”: Aristotle. THE SCHOOL of ATHENS (Raphael) 1510 -11 Pythagoras Euclid Plato Aristotle Socrates

10. 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

11. 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.

12. 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

13. 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 =14 474 011 154 664 524 427 946 373 126 085 988 481 573 677 491 474 835 889 066 354 349 131 199 152 128 (1876 Edouard Lucas) 77 digits P 13 =23562723457267347065789548996709904988477547858392600710 143020528925780432155433824984287771524270103944969186640286 44534175975063372831786222397303655396026005613602555664625 032701752803383143979023683862403317143592235664321970310172 071316352748729874740064780193958716593640108741937564905791 8549492160555646 976 (1952 Robinson) 314 digits P 1 = 6 P 2 = 28 P 3 = 496 P 4 = 8128 P 5 = ? (a 5 digit number?)

14. As you can see perfect numbers are extremely rare. There are only 41 known perfect numbers. The largest P 41 (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, M 41 is the largest known prime just as P 41 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 41 st 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!

15. Largest Prime Discovered! Monday 7 th June 2004 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 2 24 036 583 –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–digit- prime! “An award winning prime could be mere weeks away or as much as a few years away” said GIMPS founder George Woltman. 41 st 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.

16. Mersenne Primes A Mersenne number is any number of the form 2 n – 1. The first 12 of these are shown below. 2 1 – 1 = 12 2 – 1 = 3 2 3 – 1 = 7 2 4 – 1 = 15 2 5 – 1 = 31 2 6 – 1 = 63 2 7 – 1 = 127 2 8 – 1 = 255 2 9 – 1 = 511 2 10 – 1 = 10232 11 – 1 = 2047 2 12 – 1 = 4095 Early mathematicians thought that if n was prime then 2 n -1 was also prime.

17. Mersenne Primes A Mersenne number is any number of the form 2 n – 1. The first 12 of these are shown below. 2 1 – 1 = 12 2 – 1 = 3 2 3 – 1 = 7 2 4 – 1 = 15 2 5 – 1 = 31 2 6 – 1 = 63 2 7 – 1 = 127 2 8 – 1 = 255 2 9 – 1 = 511 2 10 – 1 = 10232 11 – 1 = 2047 2 12 – 1 = 4095 Early mathematicians thought that if n was prime then 2 n -1 was also prime. In 1536 Hudalricus Regius showed that 2 11 -1 = 23 x 89 and so was not prime.

18. Mersenne Primes A Mersenne Prime is any prime number of the form 2 n – 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 2 n -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. 2 2 – 1 = 3 2 3 – 1 = 7 2 5 – 1 = 31 2 7 – 1 = 127 1588 - 1644 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.

19. Mersenne Primes 2 2 – 1 = 3 2 3 – 1 = 7 2 5 – 1 = 31 2 7 – 1 = 127 1588 - 1644 2 13 – 12 17 – 1 2 19 – 1 2 31 – 1 2 127 – 1 n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, A Mersenne Prime is any number of the form 2 n – 1. were n is prime and produces a prime number.. n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257 Two of the numbers on Mersenne’s list did not generate Mersenne primes and there were others that he had missed off. Mersenne’s List Completed List 2 61 – 1 2 89 – 1 2 107 – 1 It was 1947 before all the numbers on Mersenne’s original list had been checked

20. Mersenne Primes and Perfect Numbers 2 2 – 1 = 3 2 3 – 1 = 7 2 5 – 1 = 31 2 7 – 1 = 127 1588 - 1644 2 13 – 12 17 – 1 2 19 – 1 2 31 – 1 2 127 – 1 A Mersenne Prime is any number of the form 2 n – 1. were n is prime and produces a prime number.. 2 61 – 1 2 89 – 1 2 107 – 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. n2 n -1 Perfect Number 23x? 6 37x?28 531x?496 7127x?8128

21. Mersenne Primes and Perfect Numbers 2 2 – 1 = 3 2 3 – 1 = 7 2 5 – 1 = 31 2 7 – 1 = 127 1588 - 1644 2 13 – 12 17 – 1 2 19 – 1 2 31 – 1 2 127 – 1 A Mersenne Prime is any number of the form 2 n – 1. were n is prime and produces a prime number.. 2 61 – 1 2 89 – 1 2 107 – 1 n2 n -1 Perfect Number 23x2 6 37x428 531x16496 7127x648128 Write as a power of 2 There is a formula linking a Mersenne Prime to its corresponding perfect number by multiplication. Use the table below to help you find it.

22. Mersenne Primes and Perfect Numbers 2 2 – 1 = 3 2 3 – 1 = 7 2 5 – 1 = 31 2 7 – 1 = 127 1588 - 1644 2 13 – 12 17 – 1 2 19 – 1 2 31 – 1 2 127 – 1 A Mersenne Prime is any number of the form 2 n – 1. were n is prime and produces a prime number.. 2 61 – 1 2 89 – 1 2 107 – 1 n2 n -1 Perfect Number 23x2 62121 37x4282 531x164962424 7127x6481282626 Write as a power of 2 There is a formula linking a Mersenne Prime to its corresponding perfect number by multiplication. Use the table below to help you find it.

23. Mersenne Primes and Perfect Numbers 2 2 – 1 = 3 2 3 – 1 = 7 2 5 – 1 = 31 2 7 – 1 = 127 1588 - 1644 2 13 – 12 17 – 1 2 19 – 1 2 31 – 1 2 127 – 1 A Mersenne Prime is any number of the form 2 n – 1. were n is prime and produces a prime number.. 2 61 – 1 2 89 – 1 2 107 – 1 n2 n -1 Perfect Number 23x2 62121 37x4282 531x164962424 7127x6481282626 Write as a power of 2 If 2 n -1 is a Mersenne prime then 2 n – 1 x 2 n-1 is a perfect number. Check this for the first few.

24. Mersenne Primes and Perfect Numbers 2 2 – 1 = 3 2 3 – 1 = 7 2 5 – 1 = 31 2 7 – 1 = 127 1588 - 1644 2 13 – 12 17 – 1 2 19 – 1 2 31 – 1 2 127 – 1 A Mersenne Prime is any number of the form 2 n – 1. were n is prime and produces a prime number.. 2 61 – 1 2 89 – 1 2 107 – 1 If 2 n -1 is a Mersenne prime then 2 n – 1 x 2 n-1 is a perfect number. Check this for the first few. 2 2 – 1 x 2 1 = 3 x 2 = 6 2 3 – 1 x 2 2 = 7 x 4 = 28 2 5 – 1 x 2 4 = 31 x 16 = 496 2 7 – 1 x 2 6 = 127 x 64 = 8128

25. Mersenne Primes and Perfect Numbers 2 2 – 1 = 3 2 3 – 1 = 7 2 5 – 1 = 31 2 7 – 1 = 127 1588 - 1644 2 13 – 12 17 – 1 2 19 – 1 2 31 – 1 2 127 – 1 A Mersenne Prime is any number of the form 2 n – 1. were n is prime and produces a prime number.. 2 61 – 1 2 89 – 1 2 107 – 1 NEWS FLASH 4 th September 2006 44 th Mersenne Prime Found. 2 32 582 657 – 1 has 9,808,358 digits The $100 000 prize for the world’s first 10 million digit prime is still on but you need to be quick.

26. Mersenne Primes and Perfect Numbers 2 2 – 1 = 3 2 3 – 1 = 7 2 5 – 1 = 31 2 7 – 1 = 127 1588 - 1644 2 13 – 12 17 – 1 2 19 – 1 2 31 – 1 2 127 – 1 A Mersenne Prime is any number of the form 2 n – 1. were n is prime and produces a prime number.. 2 61 – 1 2 89 – 1 2 107 – 1 Research other information about Mersenne Primes and Perfect Numbers and don’t forget to join GIMPS. G reat I nternet M ersenne P rime S earch http://www.mersenne.org/

27. Worksheet 1 FactorsA/D/PFactorsA/D/P 125 226 327 428 529 630 731 832 933 1034 1135 1236 1337 1438 1539 1640 1741 1842 1943 2044 2145 2246 2347 2448

28. n2 n -1 Perfect Number 23x 6 37x28 531x496 7127x8128 Worksheet 2 n2 n -1 Perfect Number 23x 6 37x28 531x496 7127x8128