A Lesson in the “Math + Fun!” Series Special Numbers A Lesson in the “Math + Fun!” Series 9876543210 Special Numbers

What is Special About These Numbers? Numbers in purple squares? Numbers in green squares? Circled numbers? Special Numbers

Atoms in the Universe of Numbers Are the following numbers atoms or molecules? For molecules, write down the list of atoms: 12 = 22  3 Molecule 13 = 14 = 15 = 19 = 27 = 30 = 32 = 47 = 50 = 70 = Prime number (atom) H2O Composite number (molecule) Two hydrogen atoms and one oxygen atom 13 Atom 2  7 Molecule 3  5 Molecule 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 19 Atom 33 Molecule 2  3  5 Molecule 25 Molecule 47 Atom 2  52 Molecule 2  5  7 Molecule Special Numbers

Is There a Pattern to Prime Numbers? Primes appear to be randomly distributed in this list that goes up to 620. Primes become rarer as we go higher, but there are always more primes, no matter how high we go. Special Numbers

Ulam’s Discovery 73 74 75 76 77 78 79 80 81 72 43 44 45 46 47 48 49 50 71 42 21 22 23 24 25 26 51 70 41 20 7 8 9 10 27 52 69 40 19 6 1 2 11 28 53 68 39 18 5 4 3 12 29 54 67 38 17 16 15 14 13 30 55 66 37 36 35 34 33 32 31 56 65 64 63 62 61 60 59 58 57 Primes pattern for numbers up to about 60,000; notice that primes bunch together along diagonal lines and they thin out as we move further out Stanislaw Ulam was in a boring meeting, so he started writing numbers in a spiral and discovered that prime numbers bunch together along diagonal lines Special Numbers

Ulam’s Rose Primes pattern for numbers up to 262,144. Just as water molecules bunch together to make a snowflake, prime numbers bunch together to produce Ulam’s rose. Special Numbers

Explaining Ulam’s Rose Table of numbers that is 6 columns wide shows that primes, except for 2 and 3, all fall in 2 columns 6k – 1 6k + 1 Pattern The two columns whose numbers are potentially prime form this pattern when drawn in a spiral 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 Special Numbers

Activity 1: More Number Patterns Color all boxes that contain multiples of 5 and explain the pattern that you see. Color all boxes that contain multiples of 7 and explain the pattern that you see. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 Special Numbers

Activity 2: Number Patterns in a Spiral 73 74 75 76 77 78 79 80 81 72 43 44 45 46 47 48 49 50 71 42 21 22 23 24 25 26 51 70 41 20 7 8 9 10 27 52 69 40 19 6 1 2 11 28 53 68 39 18 5 4 3 12 29 54 67 38 17 16 15 14 13 30 55 66 37 36 35 34 33 32 31 56 65 64 63 62 61 60 59 58 57 73 74 75 76 77 78 79 80 81 72 43 44 45 46 47 48 49 50 71 42 21 22 23 24 25 26 51 70 41 20 7 8 9 10 27 52 69 40 19 6 1 2 11 28 53 68 39 18 5 4 3 12 29 54 67 38 17 16 15 14 13 30 55 66 37 36 35 34 33 32 31 56 65 64 63 62 61 60 59 58 57 Color the multiples of 3. Use two different colors for odd multiples (such as 9 or 15) and for even multiples (such as 6 or 24). Color all the even numbers that are not multiples of 3 or 5. For example, 4 and 14 should be colored, but not 10 or 12. Special Numbers

Perfect Numbers A perfect number equals the sum of its divisors, except itself 6: 1 + 2 + 3 = 6 28: 1 + 2 + 4 + 7 + 14 = 28 496: 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496 An abundant number has a sum of divisors that is larger than itself 36: 1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 = 55 > 36 60: 1 + 2 + 3 + 4 + 5 + 6 + 10 + 15 + 20 + 30 = 96 > 60 100: 1 + 2 + 4 + 5 + 10 + 20 + 25 + 50 = 117 > 100 [Crub97] “Looking for Perfect Numbers” by M. Crubellier and J. Sip, Chapter 15 in [Week97]. See also, [Wago85] Wagon, “Perfect Numbers,” The Mathematical Intelligencer, Vol. 7, No. 2, pp. 66-68, 1985. A deficient number has a sum of divisors that is smaller than itself 9: 1 + 3 = 4 < 9 23: 1 < 23 128: 1 + 2 + 4 + 8 + 16 + 32 + 64 = 127 < 128 Special Numbers

Activity 3: Abundant, Deficient, or Perfect? For each of the numbers below, write down its divisors, add them up, and decide whether the number is deficient, abundant, or perfect. Number Divisors (other than the number itself) Sum of divisors Type 12 18 28 30 45 Challenge questions: Are prime numbers (for example, 2, 3, 7, 13, . . . ) abundant or deficient? Are squares of prime numbers (32 = 9, 72 = 49, . . . ) abundant or deficient? You can find powers of 2 by starting with 2 and doubling in each step. It is easy to see that 4 (divisible by 1 and 2), 8 (divisible by 1, 2, 4), and 16 (divisible by 1, 2, 4, 8) are deficient. Are all powers of 2 deficient? Special Numbers

Why Perfect Numbers Are Special Some things we know about perfect numbers There are only about a dozen perfect numbers up to 10160 All even perfect numbers end in 6 or 8 Some open questions about perfect numbers Are there an infinite set of perfect numbers? (The largest, discovered in 1997, has 120,000 digits) Are there any odd perfect numbers? (Not up to 10300) Sum of inverses of divisors of a perfect number is 1 (exclude 1/1; else 2). If there are odd perfect numbers, they must be greater than 10^300 and have at least 37 prime factors ([GUYR04] Unsolved Problems in Number Theory). Even perfect numbers are of the form 2^n(2^(n+1) – 1), with the second term prime. The largest known perfect number is 2^216,090(2^216,091 – 1). 10160 = 10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 Special Numbers

The answer is always 1089. 1089: A Very Special Number Follow these instructions: 1. Take any three digit number in which the first and last digits differ by 2 or more; e.g., 335 would be okay, but not 333 or 332. 2. Reverse the number you chose in step 1. (Example: 533) 3. You now have two numbers. Subtract the smaller number from the larger one. (Example: 533 – 335 = 198) 4. Add the answer in step 3 to the reverse of the same number. (Example: 198 + 891 = 1089) The answer is always 1089. Special Numbers

Special Numbers and Patterns Here is an amazing pattern: 12 = 1 112 = 121 1112 = 12321 11112 = 1234321 111112 = 123454321 Why is the number 37 special? 3  37 = 111 and 1 + 1 + 1 = 3 6  37 = 222 and 2 + 2 + 2 = 6 9  37 = 333 and 3 + 3 + 3 = 9 12  37 = 444 and 4 + 4 + 4 = 12 When adding or multiplying does not make a difference. You know that 2  2 = 2 + 2. But, these may be new to you: 1 1/2  3 = 1 1/2 + 3 1 1/3  4 = 1 1/3 + 4 1 1/4  5 = 1 1/4 + 5 Playing around with a number and its digits: 198 = 11 + 99 + 88 153 = 13 + 53 + 33 1634 = 14 + 64 + 34 + 44 Special Numbers

Activity 4: More Special Number Patterns Continue these patterns and find out what makes them special. 1 1 + 2 + 1 1 + 2 + 3 + 2 + 1 1 + 2 + 3 + 4 + 3 + 2 + 1 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 1 1 + 3 1 + 3 + 5 1 + 3 + 5 + 7 1 + 3 + 5 + 7 + 9 1 + 3 + 5 + 7 + 9 + 11 1 + 3 + 5 + 7 + 9 + 11 + 13 1  7 + 3 = 10 14  7 + 2 = 100 142  7 + 6 = 1000 1428  7 + 4 = 10000 14285  7 + 5 = 100000 142857  7 + 1 = 1000000 1428571  7 + 3 = 10000000 14285714  7 + 2 = 100000000 142857142  7 + 6 = 1000000000 1428571428  7 + 4 = 10000000000 1 3 + 5 7 + 9 + 11 13 + 15 + 17 + 19 21 + 23 + 25 + 27 + 29 31 + 33 + 35 + 37 + 39 + 41 43 + 45 + 47 + 49 + 51 + 53 + 55 Special Numbers

Activity 5: Special or Surprising Answers What is special about 9? 1  9 + 2 = ___ 12  9 + 3 = ____ 123  9 + 4 = _____ Can you find something special in each of the following groups? What’s special about the following? 12  483 = 5796 27  198 = 5346 39  186 = 7254 42  138 = 5796 What is special about 327? 327  1 = _____ 327  2 = _____ 327  3 = _____ Do the following multiplications: 4  1738 = _______ 4  1963 = _______ 18  297 = _______ 28  157 = _______ 48  159 = _______ Do the following multiplications: 3  51249876 = ____________ 9  16583742 = ____________ 6  32547891 = ____________ Special Numbers

Numbers as Words We can write any number as words. Here are some examples: 12 Twelve 21 Twenty-one 80 Eighty 3547 Three thousand five hundred forty-seven 0 Zero 1 One 2 Two 3 Three 4 Four 5 Five 6 Six 7 Seven 8 Eight 9 Nine 10 Ten Eight Five Four Nine One Seven Six Ten Three Two Zero Three Nine One Five Ten Seven Zero Two Four Eight Six One Two Six Ten Zero Four Five Nine Three Seven Eight Eight Four Six Ten Two Zero Five Nine One Seven Three Special Numbers

Activity 6: Numbers as Words Alpha order, from the end Evens and odds (in alpha order) Alpha order Length order 0 Zero 1 One 2 Two 3 Three 4 Four 5 Five 6 Six 7 Seven 8 Eight 9 Nine 10 Ten Eight Five Four Nine One Seven Six Ten Three Two Zero Three Nine One Five Ten Seven Zero Two Four Eight Six One Two Six Ten Zero Four Five Nine Three Seven Eight Eight Four Six Ten Two Zero Five Nine One Seven Three If we wrote these four lists from “zero” to “one thousand,” which number would appear first/last in each list? Why? What about to “one million”? Special Numbers

Activity 7: Sorting the Letters in Numbers Spell out each number and put its letters in alphabetical order (ignore hyphens and spaces). You will discover that 40 is a very special number! 0 eorz 1 eno 2 otw 3 eehrt 4 foru 5 efiv 6 isx 7 eensv 8 eghit 9 einn 10 ent 11 eeelnv 12 13 14 15 16 17 18 19 20 enttwy 21 eennottwy 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 Special Numbers

Next Lesson Math and Computers Not definite, at this point: Thursday, June 9, 2005 Math and Computers It is believed that we use decimal (base-10) numbers because humans have 10 fingers. How would we count if we had one finger on each hand? 000 001 010 011 100 101 110 111 Computers do math in base 2, because the two digits 0 and 1 that are needed are easy to represent with electronic signals or on/off switches.         Special Numbers