Steve Whealton

Prime Factor Configuration Pattern Numbers

n

pfcpn(n)

The fundamental theorem of arithmetic insists that each positive whole number can uniquely be expressed as the product of powers of prime numbers.

We will use this fundamental theorem to devise a partitioning of all natural numbers into disjoint sets, each set being identifiable by the pattern of the powers of its prime factors.

Each pattern, and thus also each set, will be given a unique numberóthe "Prime Factor Configuration Pattern Number" (pfcpn) for that pattern, and it is this pfcpn that will ultimately be used in making visual or musical patterns.

Nitty Gritty

We begin by choosing a few fairly diverse numbers that will be useful as examples:

 2364 873 2222 149 2000 6561 210 12 18 30 60 105

Next, those same numbers are shown along with their prime factors. Each factorization is given in ascending order for the component prime numbers, from left to right.

Exponents will be indicated by using the "^" symbol. For example, "2^3" will be used to signify "Two cubed," or "Two to the third power."

 2364 = 2^2 * 3^1 * 197^1 873 = 3^2 * 97^1 2222 = 2^1 * 11^1 * 101^1 149 = 149^1 2000 = 2^4 * 5^3 6561 = 3^12 210 = 2^1 * 3^1 * 5^1 * 7^1 12 = 2^2 * 3^1 18 = 2*1 * 3^2 30 = 2^1 * 3^1 * 5^1 60 = 2^2 * 3^1 * 5^1 105 = 3^1 * 5^1 * 7^1

Our task is to group the natural numbers together according to the pattern of exponents that appears when numbers are broken down into their prime factors in the manner shown above.

Below is another chart made from the same numbers. This time, the prime factors are left out, and only the exponents are shown:

 2364 2 1 1 873 2 1 2222 1 1 1 149 1 2000 4 3 6561 12 210 1 1 1 1 12 2 1 18 1 2 30 1 1 1 60 2 1 1 105 1 1 1

The idea of leaving out the prime factors and focusing only on their exponents is the key idea in this endeavor.

Several of our chosen numbers can now be seen to have the same pattern of exponents:

 2364 60 = 2 1 1 873 12 = 2 1 2222 30 105 = 1 1 1

So 60 and 2364 will end up in the same subset. 12 and 873 will go together into another subset, and 30, 105, and 2222 will belong to yet a third subset.

From Patterns to Numbers

It remains now only to decide how best to label, or enumerate, each subset, and to begin partitioning.

What we need is a chart that gives a unique number to each distinct

The first step is to create a chart that gives a unique number to each distinct prime factor configuration pattern. The pattern takes account of all of the distinct prime factors that are present. The pattern also registers to what powers those prime factors are raised. They appear in order from the smallest to the largest. I call the number given to each pattern a "prime factor configuration pattern number" (pfcpn).

First, zero and one are unique. Neither is a prime, and neither is a composite number, either. So each of them must be categorized uniquely. The most direct way to do this is to give each of them a class of its own.

 pfcpn(0) = 0 pfcpn(1) = 1

The next number, two, is a prime. The next pfcpn available is also two. So two will be the pfcpn for all primes.

 pfcpn(2) = 2 pfcpn(3) = 2 pfcpn(5) = 2 pfcpn(7) = 2 pfcpn(11) = 2 pfcpn(13) = 2 pfcpn(17) = 2 pfcpn(19) = 2 pfcpn(23) = 2 pfcpn(29) = 2

Ö and so on.

The smallest number not covered above is 4, the smallest square of a prime. The next available pfcpn available is 3. So all squares of primes will belong to the subset designated by the number 3:

 pfcpn(4) = 3 pfcpn(9) = 3 pfcpn(25) = 3 pfcpn(49) = 3 pfcpn(121) = 3 pfcpn(169) = 3 pfcpn(289) = 3 pfcpn(361) = 3 pfcpn(529) = 3 pfcpn(841) = 3

Ö etc.

Each new pattern encountered during this ascent through the integers is given the next available number:

 pfcpn(0) = 0 pfcpn(1) = 1 pfcpn(2) = 2 pfcpn(4) = 3 pfcpn(6) = 4 pfcpn(8) = 5 pfcpn(12) = 6 pfcpn(16) = 7 pfcpn(18) = 8 pfcpn(24) = 9

In the chart above, the first 10 pfcpns are indicated, along with the smallest number that exhibits each pattern. These "smallest" numbers range from 0, for pfcpn(0), up to 9, for pfcpn(24).

Notice that although both 12 and 18 are the product of one prime squared and another prime present only to the first power, the pfcpn system distinguishes between the two and gives them different numbers.

The reasoning behind this lies in the fact that with 12, the squared prime (2) is smaller than the prime that is not squared (3), whereas with 18, the prime that is squared (3) is larger than the prime that is not squared (2).

Another way of creating numbers out of prime factor configuration patterns would be to treat the two classes (as represented by 12 and 18) the same. Such a system would feature fewer distinct patterns. The difference between the two systems is very close to the difference between permutations and combinations.

The chart shown running down the left-hand side of this page shows the first 129 integers, from 0 up to 128, along with their prime factor configuration pattern numbers.

To continue the list above 128, one must keep track of each new pattern that is encountered, giving each one the next available pfcpn.

Keeping Track on the Fly

A mnemonic device is useful in working with larger numbers and their prime factor configuration patterns and numbers. The idea is to associate each configuration not so much with its pfcpn itself, but rather with the smallest number that exhibits that pfcpn. This is what the chart immediately above does.

When determining the pfcpn for, say, 2364, the first step is to factor 2364. From that factorization, the PFCP exponent pattern 2  1  1 is arrived at by then putting the prime factors, along with their exponents, in numerical order. Then the prime factors themselves are ignored, and only the exponents are considered.

The next step is to take the 2  1  1 pattern and applying it to the smallest possible set of distinct prime factors; in this case, to 2, 3, and 5. Applying the given pattern to the smallest primes will always result in the smallest number that shows the pfcpn in question:

 2^2 * 3^1 * 5^1 = 60

This tells us that 60 is the smallest number exhibiting the 2  1  1 pfcpn.

The above step, with practice, becomes easy to perform mentally. Before long, the 2  1  1 pattern evokes 60 almost automatically.

Now, you need only look up 60 in the PFCP table that you have already created. There, you will find that the pfcpn for 60 is 15. This tells you that the pfcpn for any number (such as 2364) that exhibits the 2  1  1 pattern will also be 15.

All this may seem like a lot of extra work. But the happy fact is that when you are sifting through large numbers of numbers and examining their prime factor configuration patterns, one after another, this method works. Associating a number like 2364 with 60, once you have figured out that 2  1  1 is the prime factor configuration pattern for 2364, eventually becomes rather easy; at least I find it so.

Why Bother?

So what good are these partitions and these index numbers?

Many times, in making visual or musical patterns by mathematical means, it will be necessary to squeeze a huge range of numbers down into a much smaller range. (See squeezing for more on this topic.)

Color index numbers typically lie in the ranges from 0 up to 15 or up to 255. In MIDI, the available pitches are numbered from 0 up to 127.

Although pfcpn(22) is encountered at 128, one can speculate that pfcpn(255) will not be met with until a very large number of numbers have been analyzed and their pfcpns assigned.

The process can indeed continue indefinitely. Creating a master algorithm to figure out pfcpns will require more thinking time and programming time than I have been able to devote to the task, so far.

For the time being, I keep a pfcpn chart in array form. Whenever I feel like doing some menial labor, I get out my reference books and use them to help me figure out a few more pfcpns, thus slowly enlarging my master list. Then whenever I feel like working with pfcpns, I begin with the array and build images or musical patterns around it.

Unlike the case with simple banding using primes, the visual result of using pfcpns is but subtly different from the visual result of using any of the other systems of disguising and squeezing numbers.

But once you have enjoyed and coped and struggled with as many visaual and musical patterns as I have, a reliable difference, even a subtle one, will come be to be treasured.

1

1

2

2

3

2

4

3

5

2

6

4

7

2

8

5

9

3

10

4

11

2

12

6

13

2

14

4

15

4

16

7

17

2

18

8

19

2

20

6

21

4

22

4

23

2

24

9

25

3

26

4

27

5

28

6

29

2

30

10

31

2

32

11

33

4

34

4

35

4

36

12

37

2

38

4

39

4

40

9

41

2

42

10

43

2

44

6

45

8

46

4

47

2

48

13

49

3

50

8

51

4

52

6

53

3

54

14

55

4

56

9

57

4

58

4

59

2

60

15

61

2

62

4

63

6

64

16

65

4

66

10

67

2

68

6

69

4

70

10

71

2

72

17

73

2

74

4

75

8

76

6

77

4

78

10

79

2

80

13

81

7

82

4

83

2

84

15

85

4

86

4

87

4

88

9

89

2

90

18

91

4

92

6

93

4

94

4

95

4

96

19

97

2

98

8

99

6

100

12

101

2

102

10

103

2

104

9

105

10

106

4

107

2

108

20

109

2

110

10

111

4

112

13

113

2

114

10

115

4

116

6

117

6

118

14

119

4

120

21

121

3

122

4

123

4

124

6

125

5

126

18

127

2

128

22