More Related Content
What's hot (20)
Similar to Perfect numbers and mersenne primes (20)
Recently uploaded (20)
![AVS Video Converter 12.1.5.673 Full Crack Download [Latest]](https://cdn.slidesharecdn.com/ss_thumbnails/artificialintelligence20-250528093742-69471b40-thumbnail.jpg?width=560&fit=bounds)
![Free YouTube To MP3 Converter 4.3.58.1027 Premium Crack [Latest]](https://cdn.slidesharecdn.com/ss_thumbnails/artificialintelligence11-250526151020-1862b0e3-thumbnail.jpg?width=560&fit=bounds)
![NoteBurner Spotify Music Converter 2.2.4 with Crack [Latest]](https://cdn.slidesharecdn.com/ss_thumbnails/artificialintelligence15-250528092832-e4b44d3b-thumbnail.jpg?width=560&fit=bounds)
![WinX HD Video Converter Deluxe 5.16.7.342 Crack + Key [Latest]](https://cdn.slidesharecdn.com/ss_thumbnails/artificialintelligence14-250526150454-fabc75ca-thumbnail.jpg?width=560&fit=bounds)
Perfect numbers and mersenne primes
- 2. Perfect Numbers Abundant Numbers Deficient Numbers Perfect Number: The proper divisors of a number are all its divisors excluding the number itself. Mersenne Primes
- 3. 12 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
- 4. 12 18 15 Abundant Abundant Deficient 6 1, 2, 3, 6 1 + 2 + 3 = 6 6 = 6 ? Perfect Number Perfect Number P1 = 6
- 5. The Mathematicians of Ancient Greece. Pythagoras (570 ¨C 500 BC.) Euclid of (325 ¨C 265 BC.) Archimedes (287 ¨C 212 BC.) Eratosthenes (275-192 BC.) P1 = 6 P2 = 28 P3 = 496 P4 = 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 P5
- 6. The Mathematicians of Ancient Greece. Pythagoras (570 ¨C 500 BC.) Euclid of (325 ¨C 265 BC.) Archimedes (287 ¨C 212 BC.) Eratosthenes (275-192 BC.) P1 = 6 P2 = 28 P3 = 496 P4 = 8128 P5 =? (a 5 digit number?) P5 = 33 550 336 (8 digits)
- 7. 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 P1= 6 P2= 28 P3= 496 P4 = 8128
- 8. Mersenne Primes A Mersenne number is any number of the form 2n ¨C 1 21 ¨C 1 = 1 22 ¨C 1 =3 23 ¨C 1 =7 24 ¨C 1 =15 25 ¨C 1 =31 26 ¨C 1 = 63 27 ¨C 1 = 127 28 ¨C 1 = 255 29 ¨C 1 = 511 210 ¨C 1 = 1023 211 ¨C 1 = 2047 212 ¨C 1 = 4095
- 9. Mersenne Primes A Mersenne number is any number of the form 2n ¨C 1 21 ¨C 1 = 1 22 ¨C 1 = 3 23 ¨C 1 = 7 24 ¨C 1 = 15 25 ¨C 1 = 31 26 ¨C 1 = 63 27 ¨C 1 = 127 28 ¨C 1 = 255 29 ¨C 1 = 511 210 ¨C 1 = 1023 211 ¨C 1 = 2047 212 ¨C 1 = 4095
- 10. Mersenne Primes A French monk called Marin Mersenne stated in one of his books in 1644 that for the primes: 2n ¨C 1 n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257 22 ¨C 1 = 3 23 ¨C 1 = 7 25 ¨C 1 = 31 27 ¨C 1 = 127 1588 - 1644
- 11. Mersenne Primes 22 ¨C 1 = 3 23 ¨C 1 = 7 25 ¨C 1 = 31 27 ¨C 1 = 127 1588 - 1644 213 ¨C 1 217 ¨C 1 219 ¨C 1 231 ¨C 1 2127 ¨C 1 n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257 Mersenne¡¯s List Completed List 261 ¨C 1 289 ¨C 1 2107 ¨C 1 In subsequent years various mathematicians showed that his conjecture was not correct.
- 12. Mersenne Primes and Perfect Numbers 22 ¨C 1 = 3 23 ¨C 1 = 7 25 ¨C 1 = 31 27 ¨C 1 = 127 1588 - 1644 213 ¨C 1 217 ¨C 1 219 ¨C 1 231 ¨C 1 2127 ¨C 1261 ¨C 1 289 ¨C 1 2107 ¨C 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. n 2n -1 x 2n-1 Perfect Number 2 3 x ? 6 3 7 x ? 28 5 31 x ? 496 7 127 x ? 8128
- 13. Mersenne Primes and Perfect Numbers 22 ¨C 1 = 3 23 ¨C 1 = 7 25 ¨C 1 = 31 27 ¨C 1 = 127 1588 - 1644 213 ¨C 1 217 ¨C 1 219 ¨C 1 231 ¨C 1 2127 ¨C 1261 ¨C 1 289 ¨C 1 2107 ¨C 1 n 2n -1 x 2n-1 Perfect Number 2 3 x 2 6 3 7 x 4 28 5 31 x 16 496 7 127 x 64 8128 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.
- 14. Mersenne Primes and Perfect Numbers 22 ¨C 1 = 3 23 ¨C 1 = 7 25 ¨C 1 = 31 27 ¨C 1 = 127 1588 - 1644 213 ¨C 1 217 ¨C 1 219 ¨C 1 231 ¨C 1 2127 ¨C 1261 ¨C 1 289 ¨C 1 2107 ¨C 1 n 2n -1 x 2n-1 Perfect Number 2 3 x 2 6 21 3 7 x 4 28 22 5 31 x 16 496 24 7 127 x 64 8128 26 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.
- 15. Mersenne Primes and Perfect Numbers 22 ¨C 1 = 3 23 ¨C 1 = 7 25 ¨C 1 = 31 27 ¨C 1 = 127 1588 - 1644 213 ¨C 1 217 ¨C 1 219 ¨C 1 231 ¨C 1 2127 ¨C 1261 ¨C 1 289 ¨C 1 2107 ¨C 1 If 2n -1 is a Mersenne prime then 2n ¨C 1 x 2n-1 is a perfect number. Check this for the first few. 22 ¨C 1 x 21 = 3 x 2 = 6 23 ¨C 1 x 22 = 7 x 4 = 28 25 ¨C 1 x 24 = 31 x 16 = 496 27 ¨C 1 x 26 = 127 x 64 = 8128
- 16. Mersenne Primes and Perfect Numbers 22 ¨C 1 = 3 23 ¨C 1 = 7 25 ¨C 1 = 31 27 ¨C 1 = 127 1588 - 1644 213 ¨C 1 217 ¨C 1 219 ¨C 1 231 ¨C 1 2127 ¨C 1261 ¨C 1 289 ¨C 1 2107 ¨C 1
- 17. Mersenne Primes and Perfect Numbers 22 ¨C 1 = 3 23 ¨C 1 = 7 25 ¨C 1 = 31 27 ¨C 1 = 127 1588 - 1644 213 ¨C 1 217 ¨C 1 219 ¨C 1 231 ¨C 1 2127 ¨C 1261 ¨C 1 289 ¨C 1 2107 ¨C 1 Research other information about Mersenne Primes and Perfect Numbers http://www.mersenne.org/