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March 2014

March 2014. PATTERNS vs INVARIANTS. 142857. 142857 2 × 142857 = 285714. 142857 2 × 142857 = 285714 3 × 142857 = 428571. 142857 2 × 142857 = 285714 3 × 142857 = 428571 4 × 142857 = 571428. 142857 2 × 142857 = 285714 3 × 142857 = 428571

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March 2014

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  1. March 2014

  2. PATTERNSvsINVARIANTS

  3. 142857

  4. 142857 2 × 142857 = 285714

  5. 142857 2 × 142857 = 285714 3 × 142857 = 428571

  6. 142857 2 × 142857 = 285714 3 × 142857 = 428571 4 × 142857 = 571428

  7. 142857 2 × 142857 = 285714 3 × 142857 = 428571 4 × 142857 = 571428 5 × 142857 = 714285

  8. 142857 2 × 142857 = 285714 3 × 142857 = 428571 4 × 142857 = 571428 5 × 142857 = 714285 6 × 142857 = 857142

  9. 142857 2 × 142857 = 285714 3 × 142857 = 428571 4 × 142857 = 571428 5 × 142857 = 714285 6 × 142857 = 857142 7 × 142857 = 999999

  10. Black Holes Give me any whole number (# of even digits)(# of odd digits)(total number of digits) Repeat process For Example: 19500723

  11. 19500723 # of even digits = 3 # of odd digits = 5 Total # of digits = 8 358 → 123 22221111 → 448 → 303 → 123 22222222 → 808 → 303 → 123

  12. WHY ?

  13. ooe –––> 123 oeo –––> 123 eoo –––> 123 eee –––> 303 –––> 123

  14. 201320142015 → 7512 → 134 → 123

  15. Take any 3-digit number. Rearrange it: Largest – Smallest 674: 764 – 467 = 303 303: 330 – 033 = 297 297: 972 – 279 = 693 693: 963 – 369 = 594 594: 954 – 459 = 495

  16. 738  873 – 378 = 495 954 – 459 = 495 215  512 – 125 = 396 963 – 369 = 594 954 – 459 = 495 123  321 – 123 = 198 981 – 189 = 792 972 – 279 = 693 963 – 369 = 594 954 – 459 = 495

  17. What is so special about 495? • Is there any 3–digit number does not fall into this black hole? • Is there an upper limit of number of steps to get to this black hole? • Is there any type of numbers require more steps than other types? • Is there a black hole for 2–digit numbers?

  18. What about 4-digit numbers? What about 5-digit numbers? What about 6-digit numbers? Is there a pattern?

  19. Take 2–digit numbers. 87: 87 – 78 = 9 90: 90 – 09 = 81 81 – 18 = 63 63 – 36 = 27 72 – 27 = 45 54 – 45 = 9 80: 80 – 08 = 72 72 – 27 = 45 54 – 45 = 9

  20. Is “9” a Black Hole? Is there any other Black Hole for 2–digit numbers? Which 2–digit number takes the longest to reach the Black Hole?

  21. What about 4–digit numbers? 2341: 4321 – 1234 = 3087 8730 – 0378 = 8352 8532 – 2358 = 6174 7641 – 1467 = 6174 Is 6174 the Black Hole for 4–digit numbers?

  22. What about 5–digit numbers? 12345: 54321 – 12345 = 41976 97641 – 14679 = 82962 98622 – 22689 = 75933 97533 – 33579 = 63954 96543 – 34569 = 61974 97641 – 14679 = 82962 Is 82962 the Black Hole for 5–digit numbers?

  23. Try 22220. 22220: 22220 – 02222 = 19998 99981 – 18999 = 80982 98820 – 02889 = 95931 99531 – 13599 = 85932 98532 – 23589 = 74943 97443 – 34479 = 62964 96642 – 24669 = 71973 97731 – 13779 = 83952 What happened? No Black Hole for 5–digit numbers?

  24. 22220: 22220 – 02222 = 19998 99981 – 18999 = 80982 98820 – 02889 = 95931 99531 – 13599 = 85932 98532 – 23589 = 74943 97443 – 34479 = 62964 96642 – 24669 = 71973 97731 – 13779 = 83952 98532 – 23589 = 74943 However, 74943 is different from 82962. Could it be that 5–digit numbers have more than one Black Holes?

  25. What about 6–digit, 7–digit, …, n–digit numbers?

  26. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18…

  27. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18… 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 …

  28. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 … 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 … 1 4 9 16 25 36 49 64 81…

  29. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 … 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 … 1 4 9 16 25 36 49 64 81… 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 … 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 … 1 3 7 12 19 27 37 48 61 75 91… • 8 27 64 125 216…

  30. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 … 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 … 1 4 9 16 25 36 49 64 81… 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 … 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 … 1 3 712 19 27 37 48 61 75 91… • 8 27 64 125 216… 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 … 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 … • 3 6 11 17 24 33 43 54 67 81… 1 4 15 32 65 108 175 256… • 16 81 256…

  31. Can you generalize this process and extend it? Is it same true for 5th power or 6th power or nth power?

  32. Give me any number n. Use the following process: n = odd  3 n +1 n = even  n /2

  33. Give me any number n. Use the following process: n = odd  3n+1 n = even  n/2 n=13…40, 20, 10, 5, 16, 8, 4, 2, 1 n=37…112, 56, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13,…,1 n=101…304, 152, 76, 38, 19, 58, 29, 88, 44, 22, 11,…,1

  34. Does it always go to 1? • In how many steps? • What kind of numbers require more step? • Is there a maximum number of steps? • Is there a pattern to the number of steps required?

  35. 1 = 0 2 = 1 3 = 7 4 = 2 5 = 5 6 = 8 7 = 16 8 = 3 9 = 19 10 = 6

  36. 1 = 0 11 = 14 21 = 6 2 = 1 12 = 9 22 = 15 3 = 7 13 = 9 23 = ? 4 = 2 14 = 17 24 = 10 5 = 5 15 = ? 25 = ? 6 = 8 16 = 4 26 = 10 7 = 16 17 = 12 27 = ? 8 = 3 18 = 20 28 = 18 9 = 19 19 = ? 29 = 17 10 = 6 20 = 7 30 = ?

  37. 199 is a prime Rearrange 199 into 919 (a prime). Rearrange 199 into 991 (a prime). 199 is called a “Permutable” or “Absolute” prime

  38. How many permutable primes? 1-digit: 2-digit: 3-digit:

  39. 1-digit: 4 (2, 3, 5, 7) 2-digit: 9 (11, 13, 17, 31, 37, 71, 73, 79, 97) 5 (11, 13, 17, 37, 79) 3-digit: 8 (113, 131, 199, 311, 337, 733, 919, 991) 3 (113, 199, 337) 19-digit: 1 (R19 = 11111…111) 23-digit: 1 (R23) 317-digit: 1 (R317) 1031-digit: 1 (R1031)

  40. Facts about Permutable Primes: 1. Must compose from digits 1, 3, 7, and 9 if it is not 1-digit. 2. There are no permutable primes with number of digits3 < n < 6×10175 except those that are composed all 1’s. (Not Proven) 3. There are no permutable primes with only digit 1 other than what are listed in the previous slide.(Not Proven)

  41. Facts about Permutable Primes: 4. No permutable prime exists which contains 3 different numbers of the 4 digits 1, 3, 7, 9. (For example: 731 = 17 × 43) 5. No permutable primes that are composed of 2 or more of each of the 4 digits 1, 3, 7, 9.

  42. “Unprimable” Numbers = A composite number that changes any digit and still composite. What is the smallest “unprimable” number? Example: 200

  43. For many generations, prime numbers have always had mystical and magical appeals to mathematicians. Dating back to Euclid’s days, he had already proved that there are infinitely many prime numbers. However, from what we had seen before, it is very difficult to prove even some of the simple properties about prime numbers.

  44. There are infinitely many prime numbers (Proved by Euclid about 2000 years ago). • There are infinitely many twin prime numbers (still not yet proved).

  45. There is always at least one prime number exists between n and 2n where n is a natural number larger than 1. (Proved in 1850 by Chebyshev but he used very difficult and deep mathematics in his proof. However, Erdos, the second most prolific mathematicians in history, used a very simply proof to prove this result when he was a freshman in college.)

  46. There are some numbers that are comparable to prime numbers. The great late Indian mathematician Ramanujan had considered some numbers called “highly composite” numbers.

  47. Prime numbers have the minimum number of factors…only two…1 and itself. Highly composite numbers have the maximum number of factors.

  48. Prime numbers have the minimum number of factors…only two…1 and itself. Highly composite numbers have the maximum number of factors. A composite number is Highly Composite if it has more distinct factors than any composite numbers before it.

  49. 4 has 3 distinct factors. 6 has 4 distinct factors. 8 has 4 distinct factors. 9 has 3 distinct factors. 10 has 4 distinct factors.

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